Extension of Lipschitz functions and best approximation

Costica Mustata
(original), Approximation Theory, best approximation, paper

Abstract

The aim of this paper is to present various extension results for Lipschitz functions and to put in
evidence their relevance for some best approximation problems in spaces of Lipschitz functions.

Authors

Ștefan Cobzaș

“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

Keywords

Paper coordinates

Şt. Cobzaş, C. Mustăţa,  Extension of Lipschitz functions and best approximation, Research on Theory of Allure, Approximation, Convexity and Optimization, SRIMA, Cluj-Napoca, (1999), 3-21

PDF

https://ictp.acad.ro/wp-content/uploads/2025/07/1999-Mustata-Research-on-the-Theory-of-Allure-Extension-of-Lipschitz-functions-and-best-approximation.pdf

About this paper

Journal
Publisher Name
DOI
Print ISSN
Online ISSN

google scholar link

[1] D. Andrica, C. Mustata, An abstract Korovkin type theorem and applications, Studia Univ. Babes-Bolyai, Ser. Math. 34 (1989), 41-44.
[2] R. F. Arens, J. Eells Jr., On embedding uniform and topological spaces, Pacific J. Math. 6 (1956), 397-405.
[3] Zvi Artstein, Extensions of Lipschitz selections and an application to differential inclusions, Nonlinear Anal. 16(1991), 701-704.
[4] G. Aronsson, Extension of Lipschitz functions satisfying a Lipschitz condition, Arkiv Math. 6 (1976), 551-561.
[5] N. Aronszajn, P. Panichpakdi, Extension of uniformly continuous and hyperconvex metric spaces, Pacific J. Math 6 (1956), 405-439.
[6] S. Banach, Wstep do teorii funcji rzeczwistych(Polish),[Introduction to the theory of real functions], Warszawa/Wroclaw 1951.
[7] I. Beg, M. Iqbal, Extension of linear 2-operators, Math. Montesnigri 2 (1993), 1-10.
[8] C. Bessaga, A. Pelczynski, Selected Topics in Infinite Dimensional Topology, PWN Warszawa, 1975.
[9] J. M. Borwein, M. Fabian, Characterizations of Banach spaces via convex and other locally Lipschitz functions, Acta Math. Vietnam. 22 (1997), 53-69.
[10] W. W. Breckner, Holder continuity of certain generalized convex functions, Optimization 28 (1994), 201-209.
[11] A. Bressan, A. Cortesi, Lipschitz extensions of convex-valued maps, Atti. Acad. Naz. Lincei, Rendinconti Classe Sci. Fis. Mat. Natur. (8) 80 (1986), 530-532.
[12] H. Brezis, Prolongement d’applications lipschitziennes et des semi-groupes de contractions, Seminaire Choquet: Initiation a l’analyse, 9 e anne, no. 19 (1969/70).
[13] B. M. Brown, D. Elliot, D. F. Paget, Lipschitz constant for the Bernstein polynomials of Lipschitz continuous functions, J. Approx. Theory 49 (1982), 196-199.
[14] T. Caputti, A note on the extension of Lipschitz functions, Rev. Un. Mat. Argentina 31 (1984), 122-129.
[15] E. W. Cheney, D. E. Wulbert, The existence and unicity of best approximation, Math. Scand. 24 (1969), 113-140, Corrigendum: Math. Scand. 27 (1970), 245.
[16] S. Cobzas, On the Lipschitz properties of continuous convex functions, Mathematica 21 (1979), 123-125.
[17] S. Cobzas, Lipschitz properties of convex functions, Seminar on Mathematical Analysis, Babes-Bolyai University, Faculty of Mathematics, Research Seminaries, Preprint No. 7, Cluj-Npoca 1985, 77-84.
[18] S. Cobzas, Extreme points in Banach spaces of Lipschitz functions, Mathematica 31 (1989), 25-33.
[19] S. Cobzas, I. Muntean, Continuous and locally Lipschitz convex functions, Mathematica 18 (1976), 41-51.
[20] S. Cobzas, C. Mustata, Norm preserving extension of convex Lipschitz functions, J. Approx. Theory 24 (1978), 555-564.
[21] S. Cobzas, C. Mustata, Selections associated to the metric projection, Rev. Anal. Numer. Theor. Approx. 24 (1995), 45-52.
[22] S. Cobzas, C. Mustata, Extension of bilinear operators and best approximation in 2-normed spaces, Rev. Anal. Numer. Theor. Approx. 25 (1996), 63-75
[23] S. Cobzas, C. Mustata, Extension of bilinear operators and best approximation in 2-normed spaces, Proc. 6th Workshop of the DGOR-Working Group-Multicriteria and Decision Theory, Halle 1996, A. Gopfert, J. Seelander, Chr. Tammer, Eds. Deutsche Hochschulschriften vol. 2398, Hansel-Hohenhausen Frankfurt 1997, pp. 19-29.
[24] S. Cobzas, C. Mustata, Extension of bilinear functionals and best approximation in 2-normed spaces, Studia Univ. Babes-Bolyai, Ser. Math. (in print).
[25] J. Czipser, L. Geher, Extension of functions satisfying a Lipschitz condition, Acta Math. Sci. Acad. Sci. Hungar.6 (1955), 213-220.
[26] L. Danzer, B. Grunbaum, V. Klee, Helly’s theorem and its relatives, in Convexity, Proc. Symp. Pure Appl. Math. vol. 7, Amer. Math. Soc, Providence RI 1963, pp. 101-180.
[27] V. F. Demyanov, Point derivations for Lipschitz functions and Clarke’s generalized derivative, Appl. Math. (Warsaw),24 (1997), 465-474.
[28] F. Deutsch, Linear selections for the metric projection, J. Func. Anal. 49 (1982), 269-292.
[29] F. Deutsch, A survey of the metric selections, Contemporary Mathematics vol. 18, Amer. Math. Soc., ProvidenceRI 1983, pp. 49-71.
[30] F. Deutsch, Wu Li, Sung-Ho Park, Tietze extensions and continuous selections for the metric projection, J. Approx.Theory 64 (1991), 56-68.
[31] F. Deutsch, Wu Li, Sizwe Mabizela, Helly extensions and best approximation, in Parametric Optimization and Related Topics, Approx. Optim., 3, Lang, Frankfurt am Main, 1993, pp. 107-120.
[32] M. Dorfner, The extension of Lipschitz continuous operators, Istit. Lombardo Accad. Sci. Lett. Rend. A 129 (1995),47-59.
[33] J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353-367.
[34] R. Engelking, General Topology, PWN Warszawa 1977.
[35] H. Fakhouri, Selections lineaires associees au theoreme de Hahn-Banach, J. Func. Anal. 11 (1972), 436-452.
[36] J. D. Farmer, Extreme points of the unit ball of the space of Lipschitz functions, Proc. Amer. Math. Soc. 121 (1994), 807-813.
[37] D. de Figueiredo, L. Karlovitz, On the projection into convex sets and the extension of contractions, in Proc.Conf. on Projections and Related Topics, Clemson Unic. 1967.
[38] D. de Figueiredo, L. Karlovitz, On the extension of contractions on normed spaces, in Proc. Symp. Pure Math. vol.18, part.1, Amer.Math.Soc., RI 1970.
[39] T.M. Flett, Extension of Lipschitz functions, J. London Math. Soc. 7 (1974), 604-608.
[40] C. Franchetti, Lipschitz maps on the unit ball of normed spaces, Confer. Sem. Mat. Univ. Bari vol. 202 (1985).
[41] C. Franchetti, Lipschitz maps and the geometry of the unit ball in normed spaces, Arch. Math. 46 (1986), 76-84.
[42] N. Furukama, Convexity and local Lipschitz continuity of fuzzy-valued mappings, Fuzzy Sets and Systems, 93 (1998), 113-119.
[43] S. Gahler, Linear 2-normierte Raume, Math. Nachr. 28 (1965), 1-45.
[44] B. Grunbaum, A generalization of theorems of Kirszbraun and Minty, Proc. Amer. Math. Soc. 13 (1962), 812-814.
[45] B. Grunbaum, On a theorem of Kirszbraun, Bull. Res. Council Israel 7F (1958), 129-132.
[46] B. Grunbaum, E. Zarantonello, On the extension of unniformly continuous mappings, Michigan Math. J. 15 (1968), 65-78.
[47] P. Harmand, D. Werner, W. Werner, M-ideals in Banach Spaces and Banach Algebras, Lect. Notes Math. vol. 1547 Springer-Verlag, Berlin 1993.
[48] M. Hasumi, The extension property of complex Banach spaces, Tohoku Math. J. 10 (1958), 135-142.
[49] T. Hayden, J. Wells, On the extension of Lipschitz-Holder maps of order β, J. Math. Anal. Appl. 33 (1971), 627-640.
[50] J.-B. Hiriart-Urruty, Extension of Lipschitz functions, J. Math. Anal. Appl. 77 (1980), 539-554.
[51] J.-B. Hiriart-Urruty, Extension of Lipschitz integrands and minimization of nonconvex integral functionals . Applications to the optimal resource problem in discrete time, Prob. Math. Stat. 3 (1982), 19-36.
[52] J.-B. Hiriart-Urruty, Enveloppe k-lipschitzienne d’une fonction, Rev. Math. Speciales 106 (1995-1996), 785-793.
[53] R. B. Holmes, A Course on Optimization and Best Approximation, Lect. Notes Math. vol. 257, Springer-Verlag, Berlin 1972.
[54] R. B. Holmes, Geometric Functional Analysis and its Applications, Springer-Verlag, Berlin 1975.
[55] R. B. Holmes, B. Scranton, J.D. Ward, Approximation from the space of compact operators and other M-ideals, Duke Math. J. 42 (1975(, 259-269.
[56] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rat. Mech. Anal. 123 (1993), 51-74.
[57] J. A. Johnson, Banach spaces of Lipschitz functions and vector-valued Lipschitz functions, Trans. Amer. Math. Soc. 148 (1970), 147-171.
[58] J. A. Johnson, Lipschitz spaces, Pacific J. Math. 51 (1974), 177-186.
[59] J. A. Johnson, A note on Banach spaces of Lipschitz functions, Pacific J. Math. 58 (1975), 475-482.
[60] W. B. Johnson, J. Lindenstrauss and G. Schechtman, Extensions of Lipschitz maps into Banach spaces, Israel J. Math. 54 (1986), 129-138.
[61] V. M. Kadets, Lipschitz mappings on metric spaces, Matematika, Izvestija VUZOV (1985) no.1, 30-34, (in Russian).
[62] H. Kamowitz, S. Scheinberg, Some properties of endomorphisms of Lipschitz algebras, Studia Math. 96 (1990), 61-67.
[63] J. L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952), 323-326.
[64] M. Kirszbraun,  Uber die zusammenziehenden und Lipschitzschen Transformationen, Fund. Math. 22 (1934), 77-108.
[65] S.V. Konyagin, On the level sets of Lipschitz functions, Tatra Mount. Math. Publ. 2 (1993), 51-59.
[66] A. Langenbach, Uber lokale eigenschaften von Lipschitz-Abbildungen, Math. Nachr. 194 (1998), 127-137.
[67] K. de Leeuw, Banach spaces of Lipschitz functions, Studia Math. 21 (1961), 55-66.
[68] R. Levi, M. D. Rice, The approximation of uniformly continuous mappings by Lipschitz and Holder mappings, in General Topology and its Relations to Modern Analysis and Algebra V, Prague 1981, 455-461.
[69] J. Lindensstrauss, Non-linear projections in Banach spaces, Michigan Math. J. 11 (1964), 263-287.
[70] J. Lindenstrauss, Extension of compact operators, Memoirs Amer. Math. Soc. vol. 48 (1964).
[71] S. Mabizela, On bounded 2-linear functionals, Math. Japonica 35 (1990), 51-55.
[72] S. Mabizela, The relationship between Lipschitz extensions, best approximations, and continuous selections, Quaestiones Math. 14 (1991), 261-268.
[73] L. Marco, J. A. Murillo, Locally Lipschitz and convex functions, Mathematica 38 (1996), 121-131.
[74] J. Matousek, Extension of Lipschitz mappings on metric trees, Comment. Math. Univ. Carol. 31 (1990), 99-101.
[75] J. Matousek, Note on bi-Lipschitz embeddings into normed spaces, Comment. Math. Univ. Carol. 33 (1992), 51-55.
[76] J. Matousek, On Lipschitz mappings onto square, in: The Mathematics of P. Erdos, vol. II, Springer-Verlag, Berlin 1997
[77] W. E. Mayer-Wolf, Isometries between Banach spaces of Lipschitz functions, Israel J. Math. 38 (1981), 58-74.
[78] J. A. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842.
[79] E. Mickle, On the extension of a transformation, Bull. Amer. Math. Soc. 55 (1949), 160-164.
[80] V. A. Milman, Extension of functions preserving the modulus of continuity, Mat. Zametki 61 (1997), 236-245 (in Russian).
[81] V. A. Milman, Lipschitz extensions of linearly bounded functions, Mat. Sbornik 189 (1998), 67-92 (in Russian).
[82] G. J. Minty, On the extension of Lipschitz-Holder continuous, and monotone functions, Bull. Amer. Math. Soc. 76 (1970), 334-339.
[83] C. Mustata, On some Chebyshevian subspaces of the normed space of Lipschitz functions, Rev. Anal. Numer. Teor. Aprox. 2 (1973), 81-87 (in Romanian).
[84] C. Mustata, On the unicity of the extension of continuous p-seminorms, Rev. Anal. Numer. Teor. Aprox. Vol.2 Fasc.2 (1973), 173-177 (Romanian).
[85] C. Mustata, M-ideals in metric spaces, Babes-Bolyai University Reserch Seminaries, Preprint No. 7 (1988), 65-74.
[86] C. Mustata, On the best approximation in metric spaces, Rev. Anal. Numer. Theor. Approx. 4 (1975), 45-50.
[87] C. Mustata, Norm preserving extension of starshaped Lipschitz functions, Rev. Anal. Numer. Theor. Approx. 19 (1977), 183-187.
[88] C. Mustata, Best approximation and unique extension of Lipschitz functions, J. Approx. Theory 19 (1977), 222-230.
[89] C. Mustata, A characterization of Chebyshevian subspaces of Y ?-type, Rev. Anal. Numer. Theor. Approx. 6 (1977), 51-56.
[90] C. Mustata, The extension of starshaped Lipschitz functions, Rev. Anal. Numer. Theor. Approx. 9 (1980), 93-99.
[91] C.Mustata, On the extension of Holder functions, Babes-Bolyai University Reserch Seminaries, Seminar on Functional Analysis and Numerical Methods, Preprint No. 7 (1985), 71-86.
[92] C. Mustata, On the unicity of the extension of odd Lipschitz functions, Babes-Bolyai University, Seminar on Optimization Theory, Report No. 8 (1987), 75-80.
[93] C. Mustata, Extension of Holder functions and some related problems of best approximation, Babes-Bolyai University, Faculty of Mathematics Research Seminaries, Preprint no. 7 (1991), 71-86.
[94] C. Mustata, Selections associated to McShane’s extension theorem for Lipschitz functions, Rev. Anal. Numer. Theor. Approx. 21 (1992, 135-145.
[95] C. Mustata, On the metric projection and quotient mapping, Rev. Anal. Numer. Theor. Approx. 24 (1995, 191-199.
[96] L. Nachbin, On the Hahn-Banach theorem, Anais Acad. Brasil. Ci. 21 (1949), 151-154.
[97] L. Nachbin, A theorem of Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950), 28-46.
[98] L. Nachbin, Some problems in extending and lifting continuous linear transformations, in Proc. International Symp. Linear Spaces, Jerusalem 1960, Jerusalem Acad. Press and Pergamon Press, Jerusalem and London 1961, pp. 340- 350.
[99] Nguyen Van Khue, Nguyen To Xhu, Extending locally Lipschitz maps with values in an infinite dimensional nuclear Frechet space, Bull. Acad. Polon. Sci. Ser. Math. 29 (1981), 609-616.
[100] Nguyen Van Khue, Nguyen To Xhu, Lipschitz extensions and Lipschitz retractions in metric spaces, Colloq. Math. 45 (1981), 245-250.
[101] E. Oja, On the uniqueness of the norm preserving extension of the linear functional in the Hahn-Banach theorem, Proc. Acad. Sci. Estonian SSR 33 (1984), 424-433.
[102] D. O’Keee, The tangent stars of a set, and extensions of functions in Lipschitz classes, Proc. Roy. Irish Acad. Sect. A, 97 (1997), 5-13.
[103] G. Pantelidis, Approximationstheorie fur metrische lineare Raume, Math. Ann. 184 (1969), 30-48.
[104] Sung-Ho Park, Quotient mapping, Helly-extensions, Hahn-Banach extension, Tietze extension, Lipschitz extension and best approximation, J. Korean Math. Soc. 29 (1992), 239-250.
[105] J. Partanen, J. Vaisala, Extension of bi-Lipschitz maps on compact polyhedra, Math. Scand. 72 (1993), 235-264.
[106] R. R. Phelps, Uniqueness of Hahn-Banach extension and unique best approximation, Trans. Amer. Math. Soc. 95 (1960), 238-255.
[107] B. H. Pourciau, Analysis and optimization of Lipschitz continuous mappings, J. Optim. Theory Appl. 22 (1977), 311-351.
[108] B. H. Pourciau, Hadamard’s theorem for locally Lipschitzian mappings, J. Math. Anal. Appl. 85 (1982), 279-285.
[109] B. H. Pourciau, Univalence and degree for Lipschitz continuous maps, Arch. Rat. Mech. Anal. 81 (1983), 289-299.
[110] B. H. Pourciau, Global properties of proper Lipschitzian maps, SIAM J. Math. Anal. 14 (1983), 796-799.
[111] K. Przeslawski, Lipschitz retracts, selections, and extensions, Michigan Math. J. 42 (1995), 555-571.
[112] K. Przeslawski, Lipschitz continuous selections. I. Linear selections, J. Convex Anal. 5 (1998), 249-267.
[113] S. Rolewicz, Metric Linear Spaces, PWN Warszawa 1972.
[114] S. Rolewicz, On Lipschitz projection{a geometrical approach, Ann. Univ. Maria Sklodowska-Curie 38 (1984), 135-138.
[115] S. Rolewicz, On extremal points of the unit ball in Banach spaces of Lipschitz continuous functions, J. Austral. Math. Soc. Ser. A 41 (1986), 95-98.
[116] S. Rolewicz, Generalized Asplund inequalities on Lipschitz functions, Arch. Math. 61 (1993), 484-488.
[117] S. Rolewicz, On optimal observability of Lipschitz systems, in Selected Topics in Oper. Res. and Math. Economics, Lect. Notes in Economics and Math. Systems vol 226, pp. 151-158, Springer-Verlag, Berlin 1993.
[118] S. Rolewicz, On an extension of Mazur’s theorem on Lipschitz functions, Arch. Math. 63 (1994), 535-540.
[119] S. Rolewicz, Duality and convex analysis in the absence of linear structure, Math. Japon. 44 (1996), 165-182.
[120] A. K. Roy, Extreme points and linear isometries of the Banach spaces of Lipschitz functions, Canad. J. Math. 20 (1968), 1150-1164.
[121] W. Ruess, Ein Dualkegel fur p-konvexe topologische lineare Raume, Geselschaft fur Mathematik und Datenverarbeitung Nr. 60 (1973).
[122] W. Rzymowski, Convex extension preserving Lipschitz constants, J. Math. Anal. Appl. 212 (1997), 30-37.
[123] I. Sawashima, Methods of duals in nonlinear analysis-Lipschitz duals of Banach spaces and some applications, Lect. Notes Econ. Math. Syst. vol. 419, Springer-Verlag 1975, pp.247-249.
[124] K. Schnatz, Approximationstheorie in metrischen Vektorraumen, Dissertation. Frankfurt am Main, 1985.
[125] K. Schnatz, Nonlinear duality and best approximation in metric linear spaces, J. Approx.Theory 49 (1987), 201-218.
[126] I. Schoenberg, On a theorem of Kirszbraun and Valentine, Amer. Math. Monthly 44 (1953), 620-622.
[127] S. O. Schonbeck, Extension of nonlinear contractions, Bull. Amer. Math. Soc. 72 (1966), 99-101.
[128] S. O. Schonbeck, On the extension of Lipschitz maps, Ark Math. 7 (1967-1969), 201-209.
[129] D. R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math. 13 (1963), 1387-1399.
[130] D. R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc., 111 (1964), 240-272.
[131] D. R. Shubert, A sequential method seeking the global maximum of function, SIAM J. Numer. Anal., 9 (1972), 379-388.
[132] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Ed. Academiei and Springer Verlag, Bucharest-Berlin 1970.
[133] I. Singer, Extension with larger norm and separation with double support in normed linear spaces, Bull. Austral. Math. Soc. 21 (1980), 93-106.
[134] R. Smarzewski, Extreme points of unit balls in Lipschitz functions spaces, Proc. Amer. Math. Soc. 125 (1997), 1391-1397.
[135] Ch. Stegall, Spaces of Lipschitz functions on Banach spaces, In Funct. Anal. (Essen 1991), 265-278. M. Dekker, New York 1994.
[136] V. Trenoguine, Analyse fonctionnelle, Edition Mir, Moscou 1985.
[137] S. Ustunel, Extension of Lipschitz functions on Wiener space, in Stochastic Anal. and Appl. (Powys 1995), 465-470, World Sci. 1996
[138] J. Vaisala, Bi-Lipschitz and quasi-symmetric extension properties, Ann. Acad. Sci. Fenn. Ser. I Math. 11 (1986), 239-274.
[139] J. Vaisala, Banach spaces and bi-Lipschitz maps, Studia Math. 103 (1992), 291-294.
[140] F. Valentine, A Lipschitz condition preserving extension of a vector function, Amer. J. Math. 67 (1945), 83-93.
[141] F. Valentine, On the extension of a vector function so as to preserve a Lipschitz condition, Bull. Amer. Math. Soc. 49 (1943), 100-108.
[142] F. Valentine, Contractions in non-euclidean spaces, Bull. Amer. Math. Soc. 50 (1944), 710-713.
[143] L. Waelbroeck, Closed ideals of Lipschitz functions, in Function Algebras (F. T. Birtel ed.), pp. 322-325, Scott Foresman 1966,
[144] N. Weaver, Lattices of Lipschitz functions, Pacific J. Math. 164 (1994), 179-193.
[145] N. Weaver, Isometries of noncompact Lipschitz spaces, Canad. Math. Bull. 38 (1995), 242-249.
[146] N. Weaver, Subalgebras of little Lipschitz algebras, Pacific J. Math. 173 (1996), 283-293.
[147] N. Weaver, Lipschitz algebras and derivations of von Neumann algebras, J. Func. Anal. 139 (1996), 261-300.
[148] J. H. Wells, L. R. Williams, Embeddings and Extensions in Analysis, Springer-Verlag, Berlin 1975.
[149] L. Williams, J. Wells, T. Hayden, On the extension of Lipschitz-Holder maps on Lp spaces, Studia Math. 39 (1971), 29-38.

Paper (preprint) in HTML form

1999-Mustata-Research-on-the-Theory-of-Allure-Extension-of-Lipschitz-functions-and-best-approximatio

EXTENSION OF LIPSCHITZ FUNCTIONS AND BEST APPROXIMATION

S. COBZAŞ AND C. MUSTĂŢA

The aim of this paper is to present various extension results for Lipschitz functions and to put in evidence their relevance for some best approximation problems in spaces of Lipschitz functions.

1. Extension theorems for Lipschitz functions

Let ( X , d X , d X,dX, dX,d ) be a metric space having at least two distinct points. A function f : X → R f : X → R f:X rarrRf: X \rightarrow \mathbb{R}f:X→R is called Lipschitz if there exists a number L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 such that
(1.1) | f ( x ) − f ( y ) | ≤ L d ( x , y ) (1.1) | f ( x ) − f ( y ) | ≤ L d ( x , y ) {:(1.1)|f(x)-f(y)| <= Ld(x","y):}\begin{equation*} |f(x)-f(y)| \leq L d(x, y) \tag{1.1} \end{equation*}(1.1)|f(x)−f(y)|≤Ld(x,y)
for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X.
The smallest number L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 for which (1.1) holds is called the Lipschitz norm of the function f f fff and it is denoted by ‖ f ‖ ‖ f ‖ ||f||\|f\|‖f‖. It can be calculated by the formula
(1.2) ‖ f ‖ = sup { | f ( x ) − f ( y ) | d ( x , y ) : x , y ∈ X , x ≠ y } (1.2) ‖ f ‖ = sup | f ( x ) − f ( y ) | d ( x , y ) : x , y ∈ X , x ≠ y {:(1.2)||f||=s u p{(|f(x)-f(y)|)/(d(x,y)):x,y in X,x!=y}:}\begin{equation*} \|f\|=\sup \left\{\frac{|f(x)-f(y)|}{d(x, y)}: x, y \in X, x \neq y\right\} \tag{1.2} \end{equation*}(1.2)‖f‖=sup{|f(x)−f(y)|d(x,y):x,y∈X,x≠y}
Denote by Lip X X XXX the set of all real-valued Lipschitz functions on X X XXX. With respect to the pointwise operations of addition and multiplication by scalars Lip X X XXX is a vector space and (1.2) is a seminorm on Lip X X XXX (constant functions have Lipschitz norm zero).
In order that the formula (1.2) define a proper norm we fix a point x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X (if X X XXX is a vector space, then one usually takes x 0 = 0 x 0 = 0 x_(0)=0x_{0}=0x0=0 ) and consider the space
(1.3) Lip 0 X = { f ∈ Lip X : f ( x 0 ) = 0 } (1.3) Lip 0 X = f ∈ Lip X : f x 0 = 0 {:(1.3)Lip_(0)X={f in Lip X:f(x_(0))=0}:}\begin{equation*} \operatorname{Lip}_{0} X=\left\{f \in \operatorname{Lip} X: f\left(x_{0}\right)=0\right\} \tag{1.3} \end{equation*}(1.3)Lip0X={f∈LipX:f(x0)=0}
In this case (1.2) is a norm on Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X and ( Lip 0 X , ‖ ‖ ) Lip 0 X , ‖ ‖ (Lip_(0)X,||||)\left(\operatorname{Lip}_{0} X,\| \|\right)(Lip0X,‖‖) is a Banach space, even a conjugate one. If the metric space X X XXX is bounded then the product of two Lipschitz functions is a Lipschitz function too and
(1.4) ‖ f ⋅ g ‖ ≤ ‖ f ‖ ‖ g ‖ (1.4) ‖ f ⋅ g ‖ ≤ ‖ f ‖ ‖ g ‖ {:(1.4)||f*g|| <= ||f||||g||:}\begin{equation*} \|f \cdot g\| \leq\|f\|\|g\| \tag{1.4} \end{equation*}(1.4)‖f⋅g‖≤‖f‖‖g‖
It follows that, in this case, Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X is a Banach algebra too. The properties of Banach spaces and Banach algebras of Lipschitz functions have been intensively studied in the papers of J. A. Johnson [57, 58, 59], K. de Leeuw [67], W. E. Mayer-Wolf [77, D. R. Sherbert 130, N. Weaver 144, 145, 146, 147.
As in the cases of spaces of continuous function or in the case of normed spaces an essential tool in developing the theory of spaces of Lipschitz functions is an extension theorem for Lipschitz functions. In the case of continuous functions we have Tietze's extension theorem in its scalar or vector versions (see [34] and [33]) and in the case of normed spaces we have Hahn-Banach extension theorem.
Theorem 1.1 ([78]). Let ( X , d X , d X,dX, dX,d ) be a metric space and Y Y YYY a subset of Y Y YYY. Then any real Lipschitz function on Y Y YYY admits at last one extension to X X XXX with the same Lipschitz constant.
If f ∈ Lip Y f ∈ Lip Y f in Lip Yf \in \operatorname{Lip} Yf∈LipY with Lipschitz constant L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 then two such extension are given by the formulae
(1.5) F 1 ( x ) = sup { f ( y ) − L ⋅ d ( x , y ) : y ∈ Y } , x ∈ X (1.5) F 1 ( x ) = sup { f ( y ) − L ⋅ d ( x , y ) : y ∈ Y } , x ∈ X {:(1.5)F_(1)(x)=s u p{f(y)-L*d(x","y):y in Y}","quad x in X:}\begin{equation*} F_{1}(x)=\sup \{f(y)-L \cdot d(x, y): y \in Y\}, \quad x \in X \tag{1.5} \end{equation*}(1.5)F1(x)=sup{f(y)−L⋅d(x,y):y∈Y},x∈X
and
(1.6) F 2 ( x ) = inf { f ( y ) + L ⋅ d ( x , y ) : y ∈ Y } , x ∈ X (1.6) F 2 ( x ) = inf { f ( y ) + L ⋅ d ( x , y ) : y ∈ Y } , x ∈ X {:(1.6)F_(2)(x)=i n f{f(y)+L*d(x","y):y in Y}","quad x in X:}\begin{equation*} F_{2}(x)=\inf \{f(y)+L \cdot d(x, y): y \in Y\}, \quad x \in X \tag{1.6} \end{equation*}(1.6)F2(x)=inf{f(y)+L⋅d(x,y):y∈Y},x∈X
Any other extension F F FFF with Lipschitz constant L L LLL satisfies the inequalities
(1.7) F 1 ( x ) ≤ F ( x ) ≤ F 2 ( x ) (1.7) F 1 ( x ) ≤ F ( x ) ≤ F 2 ( x ) {:(1.7)F_(1)(x) <= F(x) <= F_(2)(x):}\begin{equation*} F_{1}(x) \leq F(x) \leq F_{2}(x) \tag{1.7} \end{equation*}(1.7)F1(x)≤F(x)≤F2(x)
In particular there exists at least one extension F F FFF of f f fff verifying
(1.8) ‖ F ‖ = ‖ f ‖ (1.8) ‖ F ‖ = ‖ f ‖ {:(1.8)||F||=||f||:}\begin{equation*} \|F\|=\|f\| \tag{1.8} \end{equation*}(1.8)‖F‖=‖f‖
where, in (1.8), ‖ f ‖ ‖ f ‖ ||f||\|f\|‖f‖ is the Lipschitz norm of f f fff in Lip Y Y YYY and ‖ F ‖ ‖ F ‖ ||F||\|F\|‖F‖ is the Lipschitz norm of F F FFF in Lip X X XXX.
The problem of the extension of vector-valued functions is more complicated and delicate. The norm preserving extension is not always possible and sometimes is possible only by increasing the Lipschitz constant.
A metric space ( X , d X , d X,dX, dX,d ) is said to have the binary intersection property if any collection of mutually intersecting closed balls in X X XXX has nonvoid intersection. This property was first introduced by L. Nachbin [97] in connection with the vector version of Hahn-Banach extension property for operators. A Banach space Y Y YYY is said to have the Hahn-Banach extension property if for any normed space X X XXX and any subspace Z Z ZZZ of X X XXX every continuous linear operator A ∈ L ( Z , Y ) A ∈ L ( Z , Y ) A in L(Z,Y)A \in L(Z, Y)A∈L(Z,Y) admits a norm-preserving extension B ∈ L ( X , Y ) B ∈ L ( X , Y ) B in L(X,Y)B \in L(X, Y)B∈L(X,Y). L. Nachbin [97] proved that Y Y YYY has the Hahn-Banach extension property if and only if it has the binary intersection property and in its turn, this happens exactly when Y Y YYY is isometrically isomorphic to a space C ( T ) C ( T ) C(T)C(T)C(T), with T T TTT an extremally disconnected compact space (see J. L. Kelley [63] for the real case and M. Hasumi [48] for the complex one).
Remark that in the real case the fact that the compact T T TTT is extremally disconnected is equivalent to the fact that C ( T ) C ( T ) C(T)C(T)C(T) is a complete lattice.
The binary intersection property is also relevant in the case of the extension of Lipschitz functions, but we need a property considered by M. Kirszbraun [64] (see also F. Valentine [140]). A pair ( X , d 1 ) , ( Y , d 2 ) X , d 1 , Y , d 2 (X,d_(1)),(Y,d_(2))\left(X, d_{1}\right),\left(Y, d_{2}\right)(X,d1),(Y,d2) of metric spaces is said to have property ( K ) ( K ) (K)(K)(K) provided
(1.9) ⋂ i ∈ I B X ( x i , r i ) ≠ ∅ (1.9) ⋂ i ∈ I   B X x i , r i ≠ ∅ {:(1.9)nnn_(i in I)B_(X)(x_(i),r_(i))!=O/:}\begin{equation*} \bigcap_{i \in I} B_{X}\left(x_{i}, r_{i}\right) \neq \emptyset \tag{1.9} \end{equation*}(1.9)⋂i∈IBX(xi,ri)≠∅
implies
(1.10) ⋂ i ∈ I B Y ( y i , r i ) ≠ ∅ (1.10) ⋂ i ∈ I   B Y y i , r i ≠ ∅ {:(1.10)nnn_(i in I)B_(Y)(y_(i),r_(i))!=O/:}\begin{equation*} \bigcap_{i \in I} B_{Y}\left(y_{i}, r_{i}\right) \neq \emptyset \tag{1.10} \end{equation*}(1.10)⋂i∈IBY(yi,ri)≠∅
for all families of closed balls { B X ( x i , r i ) : i ∈ I } B X x i , r i : i ∈ I {B_(X)(x_(i),r_(i)):i in I}\left\{B_{X}\left(x_{i}, r_{i}\right): i \in I\right\}{BX(xi,ri):i∈I} and { B Y ( y i , r i ) : i ∈ I } B Y y i , r i : i ∈ I {B_(Y)(y_(i),r_(i)):i in I}\left\{B_{Y}\left(y_{i}, r_{i}\right): i \in I\right\}{BY(yi,ri):i∈I} in X X XXX and Y Y YYY, respectively, such that
(1.11) d 2 ( y i , y j ) ≤ d 1 ( x i , x j ) ( ∀ i , j ∈ I ) (1.11) d 2 y i , y j ≤ d 1 x i , x j ( ∀ i , j ∈ I ) {:(1.11)d_(2)(y_(i),y_(j)) <= d_(1)(x_(i),x_(j))quad(AA i","j in I):}\begin{equation*} d_{2}\left(y_{i}, y_{j}\right) \leq d_{1}\left(x_{i}, x_{j}\right) \quad(\forall i, j \in I) \tag{1.11} \end{equation*}(1.11)d2(yi,yj)≤d1(xi,xj)(∀i,j∈I)
One can consider also the problem of the extension of Lipschitz or Lipschitz-Hölder maps from subsets of X X XXX to Y Y YYY to the whole X X XXX.
A map f : Z → Y f : Z → Y f:Z rarr Yf: Z \rightarrow Yf:Z→Y form a subset Z Z ZZZ of X X XXX is called a Lipschitz-Hölder map of order α α alpha\alphaα if
(1.12) d 2 ( f ( x 1 ) , f ( x 2 ) ) ≤ L d 1 ( x 1 , x 2 ) α , x 1 , x 2 ∈ Z (1.12) d 2 f x 1 , f x 2 ≤ L d 1 x 1 , x 2 α , x 1 , x 2 ∈ Z {:(1.12)d_(2)(f(x_(1)),f(x_(2))) <= Ld_(1)(x_(1),x_(2))^(alpha)","quadx_(1)","x_(2)in Z:}\begin{equation*} d_{2}\left(f\left(x_{1}\right), f\left(x_{2}\right)\right) \leq L d_{1}\left(x_{1}, x_{2}\right)^{\alpha}, \quad x_{1}, x_{2} \in Z \tag{1.12} \end{equation*}(1.12)d2(f(x1),f(x2))≤Ld1(x1,x2)α,x1,x2∈Z
If α = 1 α = 1 alpha=1\alpha=1α=1 and L < 1 L < 1 L < 1L<1L<1 then f f fff is called a contraction from Z Z ZZZ to Y Y YYY. If α = 1 α = 1 alpha=1\alpha=1α=1 and L = 1 L = 1 L=1L=1L=1 then f f fff is called nonexpansive. It is called an isometry provided
(1.13) d 2 ( f ( x 1 ) , f ( x 2 ) ) = d 1 ( x 1 , x 2 ) , x 1 , x 2 ∈ Z (1.13) d 2 f x 1 , f x 2 = d 1 x 1 , x 2 , x 1 , x 2 ∈ Z {:(1.13)d_(2)(f(x_(1)),f(x_(2)))=d_(1)(x_(1),x_(2))","quadx_(1)","x_(2)in Z:}\begin{equation*} d_{2}\left(f\left(x_{1}\right), f\left(x_{2}\right)\right)=d_{1}\left(x_{1}, x_{2}\right), \quad x_{1}, x_{2} \in Z \tag{1.13} \end{equation*}(1.13)d2(f(x1),f(x2))=d1(x1,x2),x1,x2∈Z
A pair X , Y X , Y X,YX, YX,Y of metric spaces is said to have the extension property for Lip α Lip α Lip_(alpha)\operatorname{Lip}_{\alpha}Lipα-maps (nonexpansive maps, isometries) provided any Lip α − map Lip α − map Lip_(alpha)-map\operatorname{Lip}_{\alpha}-\operatorname{map}Lipα−map (nonexpansive maps, isometry) f f fff from a subset Z Z ZZZ of X X XXX
to Y Y YYY admits a Lip α Lip α Lip_(alpha)\operatorname{Lip}_{\alpha}Lipα-extension with the same Lipschitz constant (an extension which is nonexpansive, an isometry).
Supposing 0 < α ≤ 1 0 < α ≤ 1 0 < alpha <= 10<\alpha \leq 10<α≤1 then, replacing the metric d 1 d 1 d_(1)d_{1}d1 by L ⋅ d 1 α L ⋅ d 1 α L*d_(1)^(alpha)L \cdot d_{1}^{\alpha}L⋅d1α, one can suppose that f f fff is always nonexpansive, i.e. the extension problem for Lipschitz-Hölder maps of order α α alpha\alphaα with 0 < α ≤ 1 0 < α ≤ 1 0 < alpha <= 10<\alpha \leq 10<α≤1, reduces to the problem of the extension of nonexpansive maps.
Theorem 1.2 (M. Kirszbraun 64]). The pair of metric spaces ( X , d 1 X , d 1 X,d_(1)X, d_{1}X,d1 ), ( Y , d 2 Y , d 2 Y,d_(2)Y, d_{2}Y,d2 ) has the extension property for nonexpansive maps if and only if the pair ( X , Y X , Y X,YX, YX,Y ) has property ( K K KKK ).
To characterize metric spaces Y Y YYY with the extension property for nonexpansive maps for every other metric space X X XXX we need a further notion. A metric space ( Y , d 2 Y , d 2 Y,d_(2)Y, d_{2}Y,d2 ) is called metrically convex if x , y ∈ Y x , y ∈ Y x,y in Yx, y \in Yx,y∈Y and 0 < λ < 1 0 < λ < 1 0 < lambda < 10<\lambda<10<λ<1 imply the existence of a point z ∈ Y z ∈ Y z in Yz \in Yz∈Y with the property d 2 ( x , z ) = λ d 2 ( x , y ) d 2 ( x , z ) = λ d 2 ( x , y ) d_(2)(x,z)=lambdad_(2)(x,y)d_{2}(x, z)=\lambda d_{2}(x, y)d2(x,z)=λd2(x,y) and d 2 ( y , z ) = ( 1 − λ ) d 2 ( x , y ) d 2 ( y , z ) = ( 1 − λ ) d 2 ( x , y ) d_(2)(y,z)=(1-lambda)d_(2)(x,y)d_{2}(y, z)=(1-\lambda) d_{2}(x, y)d2(y,z)=(1−λ)d2(x,y). Obviously that any normed linear space is metrically convex.
Theorem 1.3 (M. Kirszbraun [64], see also [148]). A metric space Y Y YYY has the extension property for nonexpansive maps for every metric space X X XXX if and only if Y Y YYY is metrically convex and has the binary intersection property. In this case ( X , Y X , Y X,YX, YX,Y ) has the Lip α Lip α Lip_(alpha)\operatorname{Lip}_{\alpha}Lipα extension property for every metric space X X XXX and 0 < α ≤ 1 0 < α ≤ 1 0 < alpha <= 10<\alpha \leq 10<α≤1.
Theorem 1.4 ([148, 79, 126]). If H H HHH is a Hilbert space then the pair ( H , H ) ( H , H ) (H,H)(H, H)(H,H) has the extension property for nonexpansive maps and Lip α Lip α Lip_(alpha)\operatorname{Lip}_{\alpha}Lipα-maps, 0 < α ≤ 1 0 < α ≤ 1 0 < alpha <= 10<\alpha \leq 10<α≤1.
Theorem 1.5 ([148]). If H H HHH is a Hilbert space then ( H , H H , H H,HH, HH,H ) has the isometric extension property if and only if H H HHH is finite dimensional.
The following results show that, in some sense, the Hilbert space setting is the most general in the class of Banach spaces for which the contraction extension property holds.
Theorem 1.6 ([128]). If X X XXX is a strictly convex Banach space then the pair ( X , X X , X X,XX, XX,X ) has the extension property for nonexpansive maps if and only if X X XXX is a Hilbert space.
Theorem 1.7 ([128]). If X , Y X , Y X,YX, YX,Y are Banach spaces, with Y Y YYY strictly convex and of dimension at least 2 , then ( X , Y ) ( X , Y ) (X,Y)(X, Y)(X,Y) has the extension property for nonexpansive maps if and only if X X XXX and Y Y YYY are Hilbert spaces.
It can be shown that the finite dimensional space l n p , 1 < p < ∞ l n p , 1 < p < ∞ l_(n)^(p),1 < p < ool_{n}^{p}, 1<p<\inftylnp,1<p<∞, does not have the property ( K K KKK ) for n > 1 n > 1 n > 1n>1n>1 and p ≠ 2 p ≠ 2 p!=2p \neq 2p≠2. (see [148, p.50).
Other extension results for Banach valued Lipschitz functions can be found in 3, 4, 5, 39, 69, 127, 128, 141. A good account of all these problems is given in the book [148.
As in the case of norm preserving extension theorem for linear maps (Hahn-Banach theorem) it is natural to study the existence of Lipschitz extensions preserving also other properties of the function. Results of this kind are known for convex and starshaped functions.
Theorem 1.8 ([20]). Let X X XXX be a normed space and Y Y YYY a convex subset of X X XXX. Then any convex function f ∈ Lip Y f ∈ Lip Y f in Lip Yf \in \operatorname{Lip} Yf∈LipY admits a norm-preserving convex extension F ∈ Lip X F ∈ Lip X F in Lip XF \in \operatorname{Lip} XF∈LipX.
The maximal extension F 2 F 2 F_(2)F_{2}F2 given by (1.6) is convex and there exists also a minimal convex norm preserving extension F ¯ ∈ Lip X F ¯ ∈ Lip X bar(F)in Lip X\bar{F} \in \operatorname{Lip} XF¯∈LipX such that
(1.14) F ¯ ( x ) ≤ F ( x ) ≤ F 2 ( x ) , ∀ x ∈ X , (1.14) F ¯ ( x ) ≤ F ( x ) ≤ F 2 ( x ) , ∀ x ∈ X , {:(1.14) bar(F)(x) <= F(x) <= F_(2)(x)","quad AA x in X",":}\begin{equation*} \bar{F}(x) \leq F(x) \leq F_{2}(x), \quad \forall x \in X, \tag{1.14} \end{equation*}(1.14)F¯(x)≤F(x)≤F2(x),∀x∈X,
for any other convex norm preserving extension F F FFF of f f fff.
A proof of the fact that the maximal extension F 2 F 2 F_(2)F_{2}F2 of a convex Lipschitz function is convex appear also in 50] (see also 52]).
A result, similar to that proved Theorem 1.8, but for starshaped Lipschitz functions defined on starshaped subsets of Banach spaces, was proved by C. Mustăţa [87.
W. Rzymowski 122 has found the following condition in order that a function admit a convex Lipschitz extension.
Theorem 1.9. Let Ω ⊂ R n Ω ⊂ R n Omega subR^(n)\Omega \subset \mathbb{R}^{n}Ω⊂Rn be open non-empty convex and let f : bd ( Ω ) → R f : bd ( Ω ) → R f:bd(Omega)rarrRf: \operatorname{bd}(\Omega) \rightarrow \mathbb{R}f:bd(Ω)→R. The function f f fff admits a convex extension F : R n → R F : R n → R F:R^(n)rarrRF: \mathbb{R}^{n} \rightarrow \mathbb{R}F:Rn→R satisfying the Lipschitz condition with constant L L LLL if and only if the following condition is fulfilled
(1.14a) f ( z ) − f ( x ) + f ( y ) 2 ≤ L ‖ z − x + y 2 ‖ (1.14a) f ( z ) − f ( x ) + f ( y ) 2 ≤ L z − x + y 2 {:(1.14a)f(z)-(f(x)+f(y))/(2) <= L||z-(x+y)/(2)||:}\begin{equation*} f(z)-\frac{f(x)+f(y)}{2} \leq L\left\|z-\frac{x+y}{2}\right\| \tag{1.14a} \end{equation*}(1.14a)f(z)−f(x)+f(y)2≤L‖z−x+y2‖
for all x , y , z ∈ bd ( Ω ) x , y , z ∈ bd ( Ω ) x,y,z in bd(Omega)x, y, z \in \operatorname{bd}(\Omega)x,y,z∈bd(Ω).
Some extension results for Lipschitz functions on p p ppp-normed spaces were proved by W. Ruess [121]. A function ‖ ⋅ ‖ : X → R ‖ ⋅ ‖ : X → R ||*||:X rarrR\|\cdot\|: X \rightarrow \mathbb{R}‖⋅‖:X→R defined on a real vector space X X XXX is called a p p ppp-norm on X , 0 < p ≤ 1 X , 0 < p ≤ 1 X,0 < p <= 1X, 0<p \leq 1X,0<p≤1, provided it verifies the axioms
p 1 ) ‖ x ‖ ≥ 0 , ‖ x ‖ = 0 ⟺ x = 0 p 1 ) ‖ x ‖ ≥ 0 , ‖ x ‖ = 0 ⟺ x = 0 p1)||x|| >= 0,||x||=0Longleftrightarrow x=0\mathrm{p} 1)\|x\| \geq 0,\|x\|=0 \Longleftrightarrow x=0p1)‖x‖≥0,‖x‖=0⟺x=0
p2) ‖ x + y ‖ p ≤ ‖ x ‖ p + ‖ y ‖ p ‖ x + y ‖ p ≤ ‖ x ‖ p + ‖ y ‖ p ||x+y||^(p) <= ||x||^(p)+||y||^(p)\|x+y\|^{p} \leq\|x\|^{p}+\|y\|^{p}‖x+y‖p≤‖x‖p+‖y‖p
p3) ‖ λ x ‖ = | λ | ‖ x ‖ ‖ λ x ‖ = | λ | ‖ x ‖ ||lambda x||=|lambda|||x||\|\lambda x\|=|\lambda|\|x\|‖λx‖=|λ|‖x‖,
for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X and λ ∈ R λ ∈ R lambda inR\lambda \in \mathbb{R}λ∈R.
W. Ruess 121 considered as a dual for X X XXX the cone
(1.15) C X p = { h : X → R + : h ( x + y ) ≤ h ( x ) + h ( y ) and h ( λ x ) = | λ | p h ( x ) , for all x , y ∈ X and λ ∈ R } . (1.15) C X p = h : X → R + : h ( x + y ) ≤ h ( x ) + h ( y )  and  h ( λ x ) = | λ | p h ( x ) ,  for all  x , y ∈ X  and  λ ∈ R . {:[(1.15)C_(X)^(p)={h:X rarrR_(+):h(x+y) <= h(x)+h(y):}],[" and "{:h(lambda x)=|lambda|^(p)h(x)," for all "x,y in X" and "lambda inR}.]:}\begin{align*} C_{X}^{p}=\left\{h: X \rightarrow \mathbb{R}_{+}: h(x+y) \leq h(x)+h(y)\right. & \tag{1.15}\\ & \text { and } \left.h(\lambda x)=|\lambda|^{p} h(x), \text { for all } x, y \in X \text { and } \lambda \in \mathbb{R}\right\} . \end{align*}(1.15)CXp={h:X→R+:h(x+y)≤h(x)+h(y) and h(λx)=|λ|ph(x), for all x,y∈X and λ∈R}.
in Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X.
In this case one can prove the following extension result.
Theorem 1.10 ([121]). If Y Y YYY is a linear subspace of the p p ppp - normed space ( X , ‖ ⋅ ‖ X , ‖ ⋅ ‖ X,||*||X,\|\cdot\|X,‖⋅‖ ) and h ∈ C Y p h ∈ C Y p h inC_(Y)^(p)h \in C_{Y}^{p}h∈CYp then the function H : X → R + H : X → R + H:X rarrR_(+)H: X \rightarrow \mathbb{R}_{+}H:X→R+given by
(1.16) H ( x ) = inf { h ( y ) + ‖ h ‖ ⋅ ‖ x − y ‖ : y ∈ Y } (1.16) H ( x ) = inf { h ( y ) + ‖ h ‖ ⋅ ‖ x − y ‖ : y ∈ Y } {:(1.16)H(x)=i n f{h(y)+||h||*||x-y||:y in Y}:}\begin{equation*} H(x)=\inf \{h(y)+\|h\| \cdot\|x-y\|: y \in Y\} \tag{1.16} \end{equation*}(1.16)H(x)=inf{h(y)+‖h‖⋅‖x−y‖:y∈Y}
is a norm preserving extension of h h hhh in C X p C X p C_(X)^(p)C_{X}^{p}CXp, i.e.
(1.17) H ∈ C X p , H | Y = h and ‖ H ‖ = ‖ h ‖ . (1.17) H ∈ C X p , H Y = h  and  ‖ H ‖ = ‖ h ‖ . {:(1.17)H inC_(X)^(p)"," quad H|_(Y)=h quad" and "||H||=||h||.:}\begin{equation*} H \in C_{X}^{p},\left.\quad H\right|_{Y}=h \quad \text { and }\|H\|=\|h\| . \tag{1.17} \end{equation*}(1.17)H∈CXp,H|Y=h and ‖H‖=‖h‖.
where ‖ H ‖ ‖ H ‖ ||H||\|H\|‖H‖ and ‖ h ‖ ‖ h ‖ ||h||\|h\|‖h‖ stand for the Lipschitz norms (with respect to the p p ppp-norm ‖ ⋅ ‖ ‖ ⋅ ‖ ||*||\|\cdot\|‖⋅‖ ) of H H HHH and h h hhh, respectively.
These extension results can be used as a base for developing a duality theory with Lipschitz functions instead of continuous linear functional. This has been done in [123, 124, 125,
For instance K. Schnatz [124, 125] considers a metric linear space X X XXX with a translation invariant metric d d ddd and takes as dual space to X X XXX the space o o ooo - Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X formed of all odd Lipschitz functions on X X XXX, called the non-linear dual space of X X XXX, and shows that known results in the linear case as AlaogluBourbaki, Krein-Milman a.o. theorems, hold in this nonlinear case, too. Similar ideas appear in [123, but working with Lipschitz functions on a Banach space.
Another way to obtain a norm on a space of Lipschitz functions is to consider the vector space of bounded Lipschitz functions, denoted by BLip X X XXX ( X X XXX a metric space) and equip it with the norm
(1.18) ‖ f ‖ s = ‖ f ‖ L + ‖ f ‖ ∞ , (1.18) ‖ f ‖ s = ‖ f ‖ L + ‖ f ‖ ∞ , {:(1.18)||f||_(s)=||f||_(L)+||f||_(oo)",":}\begin{equation*} \|f\|_{s}=\|f\|_{L}+\|f\|_{\infty}, \tag{1.18} \end{equation*}(1.18)‖f‖s=‖f‖L+‖f‖∞,
where ‖ f ‖ L ‖ f ‖ L ||f||_(L)\|f\|_{L}‖f‖L stands for the Lipschitz norm (1.2) and ‖ f ‖ ∞ ‖ f ‖ ∞ ||f||_(oo)\|f\|_{\infty}‖f‖∞ for the uniform norm. Again BLip X X XXX is a Banach space with respect to the norm (1.18) (see [57]), and the following extension result holds true.
Theorem 1.11 ([93]). Let ( X , d X , d X,dX, dX,d ) be a metric space and Y Y YYY a subset of X X XXX. Then any f ∈ f ∈ f inf \inf∈ BLip Y Y YYY admits a norm preserving extension F ∈ F ∈ F inF \inF∈ BLip X X XXX, i.e. satisfying
(1.19) F | Y = f and ‖ F ‖ s = ‖ f ‖ s . (1.19) F Y = f  and  ‖ F ‖ s = ‖ f ‖ s . {:(1.19)F|_(Y)=f" and "||F||_(s)=||f||_(s).:}\begin{equation*} \left.F\right|_{Y}=f \text { and }\|F\|_{s}=\|f\|_{s} . \tag{1.19} \end{equation*}(1.19)F|Y=f and ‖F‖s=‖f‖s.
Two such extensions are given by
(1.20) F ¯ 1 ( x ) = { F 1 ( x ) if F 1 ( x ) ≤ ‖ f ‖ ∞ ‖ f ‖ ∞ if F 1 ( x ) > ‖ f ‖ ∞ (1.20) F ¯ 1 ( x ) = F 1 ( x )  if  F 1 ( x ) ≤ ‖ f ‖ ∞ ‖ f ‖ ∞  if  F 1 ( x ) > ‖ f ‖ ∞ {:(1.20) bar(F)_(1)(x)={[F_(1)(x)," if "quadF_(1)(x) <= ||f||_(oo)],[||f||_(oo)," if "quadF_(1)(x) > ||f||_(oo)]:}:}\bar{F}_{1}(x)= \begin{cases}F_{1}(x) & \text { if } \quad F_{1}(x) \leq\|f\|_{\infty} \tag{1.20}\\ \|f\|_{\infty} & \text { if } \quad F_{1}(x)>\|f\|_{\infty}\end{cases}(1.20)F¯1(x)={F1(x) if F1(x)≤‖f‖∞‖f‖∞ if F1(x)>‖f‖∞
and
(1.21) F ¯ 2 ( x ) = { F 2 ( x ) if F 2 ( x ) ≤ − ‖ f ‖ ∞ − ‖ f ‖ ∞ if F 2 ( x ) < − ‖ f ‖ ∞ (1.21) F ¯ 2 ( x ) = F 2 ( x )  if  F 2 ( x ) ≤ − ‖ f ‖ ∞ − ‖ f ‖ ∞  if  F 2 ( x ) < − ‖ f ‖ ∞ {:(1.21) bar(F)_(2)(x)={[F_(2)(x)," if "quadF_(2)(x) <= -||f||_(oo)],[-||f||_(oo)," if "quadF_(2)(x) < -||f||_(oo)]:}:}\bar{F}_{2}(x)= \begin{cases}F_{2}(x) & \text { if } \quad F_{2}(x) \leq-\|f\|_{\infty} \tag{1.21}\\ -\|f\|_{\infty} & \text { if } \quad F_{2}(x)<-\|f\|_{\infty}\end{cases}(1.21)F¯2(x)={F2(x) if F2(x)≤−‖f‖∞−‖f‖∞ if F2(x)<−‖f‖∞
where F 1 , F 2 F 1 , F 2 F_(1),F_(2)F_{1}, F_{2}F1,F2 are the extremal norm preserving extensions of f f fff given by (1.5) and (1.6) respectively.
Remark 1.12. The extensions F ¯ 1 , F ¯ 2 F ¯ 1 , F ¯ 2 bar(F)_(1), bar(F)_(2)\bar{F}_{1}, \bar{F}_{2}F¯1,F¯2 given by (1.20) and (1.21), preserve both the Lipschitz and uniform norms of the function f f fff, implying ‖ F ¯ 1 ‖ m = ‖ f ‖ m = ‖ F ¯ 2 ‖ m F ¯ 1 m = ‖ f ‖ m = F ¯ 2 m || bar(F)_(1)||_(m)=||f||_(m)=|| bar(F)_(2)||_(m)\left\|\bar{F}_{1}\right\|_{m}=\|f\|_{m}=\left\|\bar{F}_{2}\right\|_{m}‖F¯1‖m=‖f‖m=‖F¯2‖m, where ‖ ⋅ ‖ m ‖ ⋅ ‖ m ||*||_(m)\|\cdot\|_{m}‖⋅‖m denotes the norm on BLip X BLip X BLip X\operatorname{BLip} XBLipX, equivalent to (1.18), given by
(1.22) ‖ g ‖ m = max { ‖ g ‖ L , ‖ g ‖ ∞ } , g ∈ BLip X (1.22) ‖ g ‖ m = max ‖ g ‖ L , ‖ g ‖ ∞ , g ∈ BLip X {:(1.22)||g||_(m)=max{||g||_(L),||g||_(oo)}","quad g in BLip X:}\begin{equation*} \|g\|_{m}=\max \left\{\|g\|_{L},\|g\|_{\infty}\right\}, \quad g \in \operatorname{BLip} X \tag{1.22} \end{equation*}(1.22)‖g‖m=max{‖g‖L,‖g‖∞},g∈BLipX
The paper [93] contains also similar extension results for Hölder-Lipschitz functions of order α , 0 < α < 1 α , 0 < α < 1 alpha,0 < alpha < 1\alpha, 0< \alpha<1α,0<α<1. Furthermore, based on this extension property, one gives an algorithm for finding the global minimum of a function F ∈ Lip α X F ∈ Lip α X F inLip_(alpha)XF \in \operatorname{Lip}_{\alpha} XF∈LipαX for a compact metric space ( X , d X , d X,dX, dX,d ). One starts with the restriction of F F FFF to a subset Y Y YYY of X X XXX and one uses the norm preserving extensions of F | Y F Y F|_(Y)\left.F\right|_{Y}F|Y to X X XXX (a similar procedure was used in 131 in a particular case).
When ( X , d X , d X,dX, dX,d ) is a compact metric space, then, by the Stone-Weierstrass theorem, Lip X X XXX is dense in C ( X ) C ( X ) C(X)C(X)C(X) with respect to the uniform norm. Using this fact, some convergence results for sequences of Markov operators (linear positive operators on X X XXX which preserve the constant functions) were proved in [1].
An extension theorem for multivalued functions which are Lipschitz with respect to the HansdorffPompeiu metric was proved by A. Bressan and A. Cortesi [11. This extension does not preserve the Lipschitz constant, and the authors give an example of a multivalued Lipschitz map which does not admit extensions with the same Lipschitz constant.
Theorem 1.13 ([1]). Let H H HHH be a Hilbert space, Ω Ω Omega\OmegaΩ a subset of H H HHH and f : Ω → C ( R m ) f : Ω → C R m f:Omega rarrC(R^(m))f: \Omega \rightarrow \mathcal{C}\left(\mathbb{R}^{m}\right)f:Ω→C(Rm) a set-valued map taking values in the family C ( R m ) C R m C(R^(m))\mathcal{C}\left(\mathbb{R}^{m}\right)C(Rm) of all nonempty compact convex subsets of R m R m R^(m)\mathbb{R}^{m}Rm.
If f f fff is Lipschitz with respect to the Hansdorff-Pompeiu metric with constant L L LLL, then it admits an extension F : H → C ( R m ) F : H → C R m F:H rarrC(R^(m))F: H \rightarrow \mathcal{C}\left(\mathbb{R}^{m}\right)F:H→C(Rm) which is Lipschitz with Lipschitz constant Lm 28 / 3 28 / 3 sqrt(28//3)\sqrt{28 / 3}28/3.
Extension theorems for Lipschitz fuzzy-valued functions were proved by N. Furukama 42.

2. Applications to best approximation in spaces of Lipschitz functions

For a real normed space X X XXX a subset Y Y YYY of X X XXX and an element x ∈ X x ∈ X x in Xx \in Xx∈X put
d ( x , Y ) = inf { ‖ x − y ‖ : y ∈ Y } P Y ( x ) = { y ∈ Y : ‖ x − y ‖ = d ( x , Y ) } . d ( x , Y ) = inf { ‖ x − y ‖ : y ∈ Y } P Y ( x ) = { y ∈ Y : ‖ x − y ‖ = d ( x , Y ) } . {:[d(x","Y)=i n f{||x-y||:y in Y}],[P_(Y)(x)={y in Y:||x-y||=d(x","Y)}.]:}\begin{aligned} d(x, Y) & =\inf \{\|x-y\|: y \in Y\} \\ P_{Y}(x) & =\{y \in Y:\|x-y\|=d(x, Y)\} . \end{aligned}d(x,Y)=inf{‖x−y‖:y∈Y}PY(x)={y∈Y:‖x−y‖=d(x,Y)}.
The elements (if any) of the set P Y ( x ) P Y ( x ) P_(Y)(x)P_{Y}(x)PY(x) are called nearest points to x x xxx in Y Y YYY (or elements of best approximation). The set Y Y YYY is called proximinal if P Y ( x ) ≠ ∅ P Y ( x ) ≠ ∅ P_(Y)(x)!=O/P_{Y}(x) \neq \emptysetPY(x)≠∅ for all x ∈ X x ∈ X x in Xx \in Xx∈X, a uniqueness set if card P Y ( x ) ≤ 1 P Y ( x ) ≤ 1 P_(Y)(x) <= 1P_{Y}(x) \leq 1PY(x)≤1 for all x ∈ X x ∈ X x in Xx \in Xx∈X and Chebyshevian if card P Y ( x ) = 1 P Y ( x ) = 1 P_(Y)(x)=1P_{Y}(x)=1PY(x)=1 for all x ∈ X x ∈ X x in Xx \in Xx∈X.
If Y Y YYY is a subspace of X X XXX let
(2.1) Y ⊥ = { x ∗ ∈ X ∗ : x ∗ | Y = 0 } (2.1) Y ⊥ = x ∗ ∈ X ∗ : x ∗ Y = 0 {:(2.1)Y^(_|_)={x^(**)inX^(**):x^(**)|_(Y)=0}:}\begin{equation*} Y^{\perp}=\left\{x^{*} \in X^{*}:\left.x^{*}\right|_{Y}=0\right\} \tag{2.1} \end{equation*}(2.1)Y⊥={x∗∈X∗:x∗|Y=0}
be the annihilator space of X X XXX in X ∗ X ∗ X^(**)X^{*}X∗. R. R. Phelps [106] proved that the subspace Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is always proximinal and that it is Chebyshevian if and only if every y ∗ ∈ Y ∗ y ∗ ∈ Y ∗ y^(**)inY^(**)y^{*} \in Y^{*}y∗∈Y∗ has a unique norm preserving extension x ∗ ∈ X ∗ x ∗ ∈ X ∗ x^(**)inX^(**)x^{*} \in X^{*}x∗∈X∗.
It can be shown that similar results hold in the case of Lipschitz functions.
Let ( X , d X , d X,dX, dX,d ) be a metric space, x 0 x 0 x_(0)x_{0}x0 a fixed point in X X XXX and Y Y YYY a subset of X X XXX containing x 0 x 0 x_(0)x_{0}x0. Denote by Lip 0 X ( Lip 0 Y ) Lip 0 X Lip 0 Y Lip_(0)X(Lip_(0)Y)\operatorname{Lip}_{0} X\left(\operatorname{Lip}_{0} Y\right)Lip0X(Lip0Y) the spaces of all Lipschitz functions on X X XXX (respectively on Y Y YYY ) vanishing at x 0 ∈ Y x 0 ∈ Y x_(0)in Yx_{0} \in Yx0∈Y. Normed by (1.2) they are Banach spaces.
Put
(2.2) Y ⊥ = { F ∈ Lip 0 X : F | Y = 0 } (2.2) Y ⊥ = F ∈ Lip 0 X : F Y = 0 {:(2.2)Y^(_|_)={F inLip_(0)X:F|_(Y)=0}:}\begin{equation*} Y^{\perp}=\left\{F \in \operatorname{Lip}_{0} X:\left.F\right|_{Y}=0\right\} \tag{2.2} \end{equation*}(2.2)Y⊥={F∈Lip0X:F|Y=0}
and, for f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y, let
(2.3) E ( f ) = { F ∈ Lip 0 X : F is a norm preserving extension of f } , (2.3) E ( f ) = F ∈ Lip 0 X : F  is a norm preserving extension of  f , {:(2.3)E(f)={F inLip_(0)X:F" is a norm preserving extension of "f}",":}\begin{equation*} E(f)=\left\{F \in \operatorname{Lip}_{0} X: F \text { is a norm preserving extension of } f\right\}, \tag{2.3} \end{equation*}(2.3)E(f)={F∈Lip0X:F is a norm preserving extension of f},
i.e.
F ∈ E ( f ) ⟺ F | Y = f and ‖ F ‖ X = ‖ f ‖ Y . F ∈ E ( f ) ⟺ F Y = f  and  ‖ F ‖ X = ‖ f ‖ Y .  F in E(f)Longleftrightarrow F|_(Y)=f quad" and "quad||F||_(X)=||f||_(Y)". "\left.F \in E(f) \Longleftrightarrow F\right|_{Y}=f \quad \text { and } \quad\|F\|_{X}=\|f\|_{Y} \text {. }F∈E(f)⟺F|Y=f and ‖F‖X=‖f‖Y. 
By the extension theorem (Theorem 1.1) the set E ( f ) E ( f ) E(f)E(f)E(f) is non-empty for any f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y.
Theorem 2.1 ([83]). Let ( X , d ) , x 0 , Y ( X , d ) , x 0 , Y (X,d),x_(0),Y(X, d), x_{0}, Y(X,d),x0,Y be as above and Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ be defined by (2.2)
1 ∘ 1 ∘ 1^(@)1^{\circ}1∘ The space Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is always proximinal in Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X and for every F ∈ Lip 0 X F ∈ Lip 0 X F inLip_(0)XF \in \operatorname{Lip}_{0} XF∈Lip0X
(2.4) d ( F , Y ⊥ ) = ‖ F | Y ‖ (2.4) d F , Y ⊥ = F Y {:(2.4)d(F,Y^(_|_))=||F|_(Y)||:}\begin{equation*} d\left(F, Y^{\perp}\right)=\left\|\left.F\right|_{Y}\right\| \tag{2.4} \end{equation*}(2.4)d(F,Y⊥)=‖F|Y‖
and
(2.5) P Y ⊥ ( F ) = F − E ( F | Y ) (2.5) P Y ⊥ ( F ) = F − E F Y {:(2.5)P_(Y^(_|_))(F)=F-E(F|_(Y)):}\begin{equation*} P_{Y^{\perp}}(F)=F-E\left(\left.F\right|_{Y}\right) \tag{2.5} \end{equation*}(2.5)PY⊥(F)=F−E(F|Y)
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ The space Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is Chebyshevian in Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X if and only if every f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y has a unique norm preserving extension F ∈ Lip 0 X F ∈ Lip 0 X F inLip_(0)XF \in \operatorname{Lip}_{0} XF∈Lip0X.
Proof. For F ∈ Lip 0 X F ∈ Lip 0 X F inLip_(0)XF \in \operatorname{Lip}_{0} XF∈Lip0X and arbitrary G ∈ Y ⊥ G ∈ Y ⊥ G inY^(_|_)G \in Y^{\perp}G∈Y⊥ we have
‖ F | Y ‖ = sup { | F ( x ) − F ( y ) | d ( x , y ) : x , y ∈ Y , x ≠ y } = sup { | ( F − G ) ( x ) − ( F − G ) ( y ) | d ( x , y ) : x , y ∈ Y ; x ≠ y } ≤ ≤ sup { | ( F − G ) ( x ) − ( F − G ) ( y ) | d ( x , y ) : x , y ∈ X ; x ≠ y } = ‖ F − G ‖ F Y = sup | F ( x ) − F ( y ) | d ( x , y ) : x , y ∈ Y , x ≠ y = sup | ( F − G ) ( x ) − ( F − G ) ( y ) | d ( x , y ) : x , y ∈ Y ; x ≠ y ≤ ≤ sup | ( F − G ) ( x ) − ( F − G ) ( y ) | d ( x , y ) : x , y ∈ X ; x ≠ y = ‖ F − G ‖ {:[||F|_(Y)||=s u p{(|F(x)-F(y)|)/(d(x,y)):x,y in Y,x!=y}],[=s u p{(|(F-G)(x)-(F-G)(y)|)/(d(x,y)):x,y in Y;x!=y} <= ],[ <= s u p{(|(F-G)(x)-(F-G)(y)|)/(d(x,y)):x,y in X;x!=y}=||F-G||]:}\begin{aligned} \left\|\left.F\right|_{Y}\right\| & =\sup \left\{\frac{|F(x)-F(y)|}{d(x, y)}: x, y \in Y, x \neq y\right\} \\ & =\sup \left\{\frac{|(F-G)(x)-(F-G)(y)|}{d(x, y)}: x, y \in Y ; x \neq y\right\} \leq \\ & \leq \sup \left\{\frac{|(F-G)(x)-(F-G)(y)|}{d(x, y)}: x, y \in X ; x \neq y\right\}=\|F-G\| \end{aligned}‖F|Y‖=sup{|F(x)−F(y)|d(x,y):x,y∈Y,x≠y}=sup{|(F−G)(x)−(F−G)(y)|d(x,y):x,y∈Y;x≠y}≤≤sup{|(F−G)(x)−(F−G)(y)|d(x,y):x,y∈X;x≠y}=‖F−G‖
implying
‖ F | Y ‖ ≤ inf { ‖ F − G ‖ : G ∈ Y ⊥ } = d ( F , Y ⊥ ) F Y ≤ inf ‖ F − G ‖ : G ∈ Y ⊥ = d F , Y ⊥ ||F|_(Y)|| <= i n f{||F-G||:G inY^(_|_)}=d(F,Y^(_|_))\left\|\left.F\right|_{Y}\right\| \leq \inf \left\{\|F-G\|: G \in Y^{\perp}\right\}=d\left(F, Y^{\perp}\right)‖F|Y‖≤inf{‖F−G‖:G∈Y⊥}=d(F,Y⊥)
By Theorem 1.1 there exists G ∈ Lip 0 X G ∈ Lip 0 X G inLip_(0)XG \in \operatorname{Lip}_{0} XG∈Lip0X such that G | Y = F | Y G Y = F Y G|_(Y)=F|_(Y)\left.G\right|_{Y}=\left.F\right|_{Y}G|Y=F|Y and ‖ G ‖ = ‖ F | Y ‖ ‖ G ‖ = F Y ||G||=||F|_(Y)||\|G\|=\left\|\left.F\right|_{Y}\right\|‖G‖=‖F|Y‖. It follows F − G ∈ Y ⊥ F − G ∈ Y ⊥ F-G inY^(_|_)F-G \in Y^{\perp}F−G∈Y⊥ and
d ( F , Y ⊥ ) ≤ ‖ F − ( F − G ) ‖ = ‖ G ‖ = ‖ F | Y ‖ , d F , Y ⊥ ≤ ‖ F − ( F − G ) ‖ = ‖ G ‖ = F Y , d(F,Y^(_|_)) <= ||F-(F-G)||=||G||=||F|_(Y)||,d\left(F, Y^{\perp}\right) \leq\|F-(F-G)\|=\|G\|=\left\|\left.F\right|_{Y}\right\|,d(F,Y⊥)≤‖F−(F−G)‖=‖G‖=‖F|Y‖,
showing that (2.4) holds.
Also G ∈ Y ⊥ G ∈ Y ⊥ G inY^(_|_)G \in Y^{\perp}G∈Y⊥ is a nearest point to F F FFF in Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ iff
‖ F − G ‖ = d ( F , Y ⊥ ) = ‖ F | Y ‖ ‖ F − G ‖ = d F , Y ⊥ = F Y ||F-G||=d(F,Y^(_|_))=||F|_(Y)||\|F-G\|=d\left(F, Y^{\perp}\right)=\left\|\left.F\right|_{Y}\right\|‖F−G‖=d(F,Y⊥)=‖F|Y‖
Since ( F − G ) | Y = F | Y ( F − G ) Y = F Y (F-G)|_(Y)=F|_(Y)\left.(F-G)\right|_{Y}=\left.F\right|_{Y}(F−G)|Y=F|Y we have F − G ∈ E ( F | Y ) F − G ∈ E F Y F-G in E(F|_(Y))F-G \in E\left(\left.F\right|_{Y}\right)F−G∈E(F|Y) which is equivalent to G ∈ F − E ( F | Y ) G ∈ F − E F Y G in F-E(F|_(Y))G \in F-E\left(\left.F\right|_{Y}\right)G∈F−E(F|Y).
Therefore
G ∈ P Y ⊥ ( F ) ⟺ G ∈ F − E ( F | Y ) G ∈ P Y ⊥ ( F ) ⟺ G ∈ F − E F Y G inP_(Y^(_|_))(F)Longleftrightarrow G in F-E(F|_(Y))G \in P_{Y^{\perp}}(F) \Longleftrightarrow G \in F-E\left(\left.F\right|_{Y}\right)G∈PY⊥(F)⟺G∈F−E(F|Y)
proving the formula (2.5).
The second assertion of the theorem is an immediate consequence of this formula.
By Theorem 1.1, any norm preserving extension of F | Y F Y F|_(Y)\left.F\right|_{Y}F|Y is contained between the extremal extensions F 1 , F 2 F 1 , F 2 F_(1),F_(2)F_{1}, F_{2}F1,F2 given by (1.5) and (1.6), respectively, it follows that E ( F | Y ) E F Y E(F|_(Y))E\left(\left.F\right|_{Y}\right)E(F|Y) is a singleton if and only if
sup { F | Y ( y ) − ‖ F | Y ‖ d ( x , y ) : y ∈ Y } = inf { F | Y ( y ) + ‖ F | Y ‖ d ( x , y ) : y ∈ Y } sup F Y ( y ) − F Y d ( x , y ) : y ∈ Y = inf F Y ( y ) + F Y d ( x , y ) : y ∈ Y {:[ s u p{F|_(Y)(y)-||F|_(Y)||d(x,y):y in Y}=],[ i n f{F|_(Y)(y)+||F|_(Y)||d(x,y):y in Y}]:}\begin{aligned} & \sup \left\{\left.F\right|_{Y}(y)-\left\|\left.F\right|_{Y}\right\| d(x, y): y \in Y\right\}= \\ & \inf \left\{\left.F\right|_{Y}(y)+\left\|\left.F\right|_{Y}\right\| d(x, y): y \in Y\right\} \end{aligned}sup{F|Y(y)−‖F|Y‖d(x,y):y∈Y}=inf{F|Y(y)+‖F|Y‖d(x,y):y∈Y}
for all x ∈ X x ∈ X x in Xx \in Xx∈X.
Since
inf ( F | Y ) ( Y ) + ‖ F | Y ‖ d ( x , y ) ≤ sup ( F | Y ) ( Y ) − ‖ F | Y ‖ d ( x , Y ) inf F Y ( Y ) + F Y d ( x , y ) ≤ sup F Y ( Y ) − F Y d ( x , Y ) i n f(F|_(Y))(Y)+||F|_(Y)||d(x,y) <= s u p(F|_(Y))(Y)-||F|_(Y)||d(x,Y)\inf \left(\left.F\right|_{Y}\right)(Y)+\left\|\left.F\right|_{Y}\right\| d(x, y) \leq \sup \left(\left.F\right|_{Y}\right)(Y)-\left\|\left.F\right|_{Y}\right\| d(x, Y)inf(F|Y)(Y)+‖F|Y‖d(x,y)≤sup(F|Y)(Y)−‖F|Y‖d(x,Y)
we have
(2.6) d ( x , Y ) ≤ sup ( F | Y ) ( Y ) − inf ( F | Y ) ( Y ) 2 ‖ F | Y ‖ (2.6) d ( x , Y ) ≤ sup F Y ( Y ) − inf F Y ( Y ) 2 F Y {:(2.6)d(x","Y) <= (s u p(F|_(Y))(Y)-i n f(F|_(Y))(Y))/(2||F|_(Y)||):}\begin{equation*} d(x, Y) \leq \frac{\sup \left(\left.F\right|_{Y}\right)(Y)-\inf \left(\left.F\right|_{Y}\right)(Y)}{2\left\|\left.F\right|_{Y}\right\|} \tag{2.6} \end{equation*}(2.6)d(x,Y)≤sup(F|Y)(Y)−inf(F|Y)(Y)2‖F|Y‖
for every x ∈ X x ∈ X x in Xx \in Xx∈X and every F ∈ Lip 0 X ∖ Y ⊥ F ∈ Lip 0 X ∖ Y ⊥ F inLip_(0)X\\Y^(_|_)F \in \operatorname{Lip}_{0} X \backslash Y^{\perp}F∈Lip0X∖Y⊥.
Using this inequality one can give conditions on Y Y YYY in order that its annihilator Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ be Chebyshevian.
Proposition 2.2. Let ( X , d X , d X,dX, dX,d ) be a metric space and Y Y YYY a subset of X X XXX containing the distinguished point x 0 x 0 x_(0)x_{0}x0.
1 ∘ 1 ∘ 1^(@)1^{\circ}1∘ If Y ¯ = X Y ¯ = X bar(Y)=X\bar{Y}=XY¯=X, then Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is Chebyshevian in Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ If Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is Chebyshevian and Y Y YYY contains at least one accumulation point, then Y ¯ = X Y ¯ = X bar(Y)=X\bar{Y}=XY¯=X.
Similar results hold in the case of the extension of convex Lipschitz functions.
By Theorem 1.8 every convex function f ∈ Lip Y f ∈ Lip Y f in Lip Yf \in \operatorname{Lip} Yf∈LipY admits a convex norm preserving extension F ∈ Lip X F ∈ Lip X F in Lip XF \in \operatorname{Lip} XF∈LipX. The minimal extension F 1 F 1 F_(1)F_{1}F1 given by (1.5) is convex and there exists a maximal extension F ¯ F ¯ bar(F)\bar{F}F¯ too.
Now, if 0 ∈ Y 0 ∈ Y 0in Y0 \in Y0∈Y is the fixed point, then put
K Y = { f ∈ Lip 0 Y : f is convex on Y } K Y = f ∈ Lip 0 Y : f  is convex on  Y K_(Y)={f inLip_(0)Y:f" is convex on "Y}K_{Y}=\left\{f \in \operatorname{Lip}_{0} Y: f \text { is convex on } Y\right\}KY={f∈Lip0Y:f is convex on Y}
It follows that K Y K Y K_(Y)K_{Y}KY is a convex cone and let
X c = K X − K X X c = K X − K X X_(c)=K_(X)-K_(X)X_{c}=K_{X}-K_{X}Xc=KX−KX
be the linear space generated by the cone
K X = { F ∈ Lip 0 X : F is convex on X } K X = F ∈ Lip 0 X : F  is convex on  X K_(X)={F inLip_(0)X:F" is convex on "X}K_{X}=\left\{F \in \operatorname{Lip}_{0} X: F \text { is convex on } X\right\}KX={F∈Lip0X:F is convex on X}
Let also
Y c = K Y − K Y Y c = K Y − K Y Y_(c)=K_(Y)-K_(Y)Y_{c}=K_{Y}-K_{Y}Yc=KY−KY
and
Y c ⊥ = { F ∈ X c : F | Y = 0 } Y c ⊥ = F ∈ X c : F Y = 0 Y_(c)^(_|_)={F inX_(c):F|_(Y)=0}Y_{c}^{\perp}=\left\{F \in X_{c}:\left.F\right|_{Y}=0\right\}Yc⊥={F∈Xc:F|Y=0}
Theorem 2.3 ([20]). 1 ∘ 1 ∘ 1^(@)1^{\circ}1∘ If F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX, then
‖ F | Y ‖ = d ( F , Y c ⊥ ) F Y = d F , Y c ⊥ ||F|_(Y)||=d(F,Y_(c)^(_|_))\left\|\left.F\right|_{Y}\right\|=d\left(F, Y_{c}^{\perp}\right)‖F|Y‖=d(F,Yc⊥)
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ The space Y c ⊥ Y c ⊥ Y_(c)^(_|_)Y_{c}^{\perp}Yc⊥ is K X K X K_(X)K_{X}KX-proximinal and, for F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX, a function G ∈ Y c ⊥ G ∈ Y c ⊥ G inY_(c)^(_|_)G \in Y_{c}^{\perp}G∈Yc⊥ is a nearest point to F F FFF in Y c ⊥ Y c ⊥ Y_(c)^(_|_)Y_{c}^{\perp}Yc⊥ if and only if G = F − H G = F − H G=F-HG=F-HG=F−H where H H HHH is a convex norm preserving extension of F | Y F Y F|_(Y)\left.F\right|_{Y}F|Y.
3 ∘ 3 ∘ 3^(@)3^{\circ}3∘ The space Y c ⊥ Y c ⊥ Y_(c)^(_|_)Y_{c}^{\perp}Yc⊥ is K X K X K_(X)K_{X}KX-Chebyshevian if and only if every f ∈ K Y f ∈ K Y f inK_(Y)f \in K_{Y}f∈KY has a unique convex norm preserving extension to X X XXX.
Similar results hold for starshaped Lipschitz functions (see [90]).

References

[1] D. Andrica, C. Mustăţa, An abstract Korovkin type theorem and applications, Studia Univ. Babes-Bolyai, Ser. Math. 34 (1989), 41-44.
[2] R. F. Arens, J. Eells Jr., On embedding uniform and topological spaces, Pacific J. Math. 6 (1956), 397-405.
[3] Zvi Artstein, Extensions of Lipschitz selections and an application to differential inclusions, Nonlinear Anal. 16 (1991), 701-704.
[4] G. Aronsson, Extension of Lipschitz functions satisfying a Lipschitz condition, Arkiv Math. 6 (1976), 551-561.
[5] N. Aronszajn, P. Panichpakdi, Extension of uniformly continuous and hyperconvex metric spaces, Pacific J. Math 6 (1956), 405-439.
[6] S. Banach, Wstep do teorii funcji rzeczwistych(Polish),[Introduction to the theory of real functions], Warszawa/Wroclaw 1951.
[7] I. Beg, M. Iqbal, Extension of linear 2-operators, Math. Montesnigri 2 (1993), 1-10.
[8] C. Bessaga, A. Pelczynski, Selected Topics in Infinite Dimensional Topology, PWN Warszawa, 1975.
[9] J. M. Borwein, M. Fabian, Characterizations of Banach spaces via convex and other locally Lipschitz functions, Acta Math. Vietnam. 22 (1997), 53-69.
[10] W. W. Breckner, Hölder continuity of certain generalized convex functions, Optimization 28 (1994), 201-209.
[11] A. Bressan, A. Cortesi, Lipschitz extensions of convex-valued maps, Atti. Acad. Naz. Lincei, Rendinconti Classe Sci. Fis. Mat. Natur. (8) 80 (1986), 530-532.
[12] H. Brezis, Prolongement d'applications lipschitziennes et des semi-groupes de contractions, Séminaire Choquet: Initiation à l'analyse, 9 e anné, no. 19 (1969/70).
[13] B. M. Brown, D. Elliot, D. F. Paget, Lipschitz constant for the Bernstein polynomials of Lipschitz continuous functions, J. Approx. Theory 49 (1982), 196-199.
[14] T. Caputti, A note on the extension of Lipschitz functions, Rev. Un. Mat. Argentina 31 (1984), 122-129.
[15] E. W. Cheney, D. E. Wulbert, The existence and unicity of best approximation, Math. Scand. 24 (1969), 113-140, Corrigendum: Math. Scand. 27 (1970), 245.
[16] S. Cobzaş, On the Lipschitz properties of continuous convex functions, Mathematica 21 (1979), 123-125.
[17] S. Cobzaş, Lipschitz properties of convex functions, Seminar on Mathematical Analysis, Babes-Bolyai University, Faculty of Mathematics, Research Seminaries, Preprint No. 7, Cluj-Npoca 1985, 77-84.
[18] S. Cobzaş, Extreme points in Banach spaces of Lipschitz functions, Mathematica 31 (1989), 25-33.
[19] S. Cobzaş, I. Muntean, Continuous and locally Lipschitz convex functions, Mathematica 18 (1976), 41-51.
[20] S. Cobzaş, C. Mustăţa, Norm preserving extension of convex Lipschitz functions, J. Approx. Theory 24 (1978), 555-564.
[21] S. Cobzaş, C. Mustăţa, Selections associated to the metric projection, Rev. Anal. Numér. Théor. Approx. 24 (1995), 45-52.
[22] S. Cobzaş, C. Mustăţa, Extension of bilinear operators and best approximation in 2-normed spaces, Rev. Anal. Numér. Théor. Approx. 25 (1996), 63-75
[23] S. Cobzaş, C. Mustăţa, Extension of bilinear operators and best approximation in 2-normed spaces, Proc. 6th Workshop of the DGOR-Working Group-Multicriteria and Decision Theory, Halle 1996, A. Göpfert, J. Seeländer, Chr. Tammer, Eds. Deutsche Hochschulschriften vol. 2398, H"ansel-Hohenhausen Frankfurt 1997, pp. 19-29.
[24] S. Cobzaş, C. Mustăţa, Extension of bilinear functionals and best approximation in 2-normed spaces, Studia Univ. Babes-Bolyai, Ser. Math. (in print).
[25] J. Czipser, L. Gehér, Extension of functions satisfying a Lipschitz condition, Acta Math. Sci. Acad. Sci. Hungar. 6 (1955), 213-220.
[26] L. Danzer, B. Grünbaum, V. Klee, Helly's theorem and its relatives, in Convexity, Proc. Symp. Pure Appl. Math. vol. 7, Amer. Math. Soc, Providence RI 1963, pp. 101-180.
[27] V. F. Demyanov, Point derivations for Lipschitz functions and Clarke's generalized derivative, Appl. Math. (Warsaw), 24 (1997), 465-474.
[28] F. Deutsch, Linear selections for the metric projection, J. Func. Anal. 49 (1982), 269-292.
[29] F. Deutsch, A survey of the metric selections, Contemporary Mathematics vol. 18, Amer. Math. Soc., Providence RI 1983, pp. 49-71.
[30] F. Deutsch, Wu Li, Sung-Ho Park, Tietze extensions and continuous selections for the metric projection, J. Approx. Theory 64 (1991), 56-68.
[31] F. Deutsch, Wu Li, Sizwe Mabizela, Helly extensions and best approximation, in Parametric Optimization and Related Topics, Approx. Optim., 3, Lang, Frankfurt am Main, 1993, pp. 107-120.
[32] M. Dorfner, The extension of Lipschitz continuous operators, Istit. Lombardo Accad. Sci. Lett. Rend. A 129 (1995), 47-59.
[33] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367.
[34] R. Engelking, General Topology, PWN Warszawa 1977.
[35] H. Fakhouri, Séléctions linéaires associées au théorème de Hahn-Banach, J. Func. Anal. 11 (1972), 436-452.
[36] J. D. Farmer, Extreme points of the unit ball of the space of Lipschitz functions, Proc. Amer. Math. Soc. 121 (1994), 807-813.
[37] D. de Figueiredo, L. Karlovitz, On the projection into convex sets and the extension of contractions, in Proc.Conf. on Projections and Related Topics, Clemson Unic. 1967.
[38] D. de Figueiredo, L. Karlovitz, On the extension of contractions on normed spaces, in Proc. Symp. Pure Math. vol.18, part.1, Amer.Math.Soc., RI 1970.
[39] T.M. Flett, Extension of Lipschitz functions, J. London Math. Soc. 7 (1974), 604-608.
[40] C. Franchetti, Lipschitz maps on the unit ball of normed spaces, Confer. Sem. Mat. Univ. Bari vol. 202 (1985).
[41] C. Franchetti, Lipschitz maps and the geometry of the unit ball in normed spaces, Arch. Math. 46 (1986), 76-84.
[42] N. Furukama, Convexity and local Lipschitz continuity of fuzzy-valued mappings, Fuzzy Sets and Systems, 93 (1998), 113-119.
[43] S. Gähler, Linear 2-normierte Räume, Math. Nachr. 28 (1965), 1-45.
[44] B. Grünbaum, A generalization of theorems of Kirszbraun and Minty, Proc. Amer. Math. Soc. 13 (1962), 812-814.
[45] B. Grünbaum, On a theorem of Kirszbraun, Bull. Res. Council Israel 7F (1958), 129-132.
[46] B. Grünbaum, E. Zarantonello, On the extension of unniformly continuous mappings, Michigan Math. J. 15 (1968), 65-78.
[47] P. Harmand, D. Werner, W. Werner, M-ideals in Banach Spaces and Banach Algebras, Lect. Notes Math. vol. 1547 Springer-Verlag, Berlin 1993.
[48] M. Hasumi, The extension property of complex Banach spaces, Tohoku Math. J. 10 (1958), 135-142.
[49] T. Hayden, J. Wells, On the extension of Lipschitz-Hölder maps of order β β beta\betaβ, J. Math. Anal. Appl. 33 (1971), 627-640.
[50] J.-B. Hiriart-Urruty, Extension of Lipschitz functions, J. Math. Anal. Appl. 77 (1980), 539-554.
[51] J.-B. Hiriart-Urruty, Extension of Lipschitz integrands and minimization of nonconvex integral functionals . Applications to the optimal resource problem in discrete time, Prob. Math. Stat. 3 (1982), 19-36.
[52] J.-B. Hiriart-Urruty, Enveloppe k-lipschitzienne d'une fonction, Rev. Math. Spéciales 106 (1995-1996), 785-793.
[53] R. B. Holmes, A Course on Optimization and Best Approximation, Lect. Notes Math. vol. 257, Springer-Verlag, Berlin 1972.
[54] R. B. Holmes, Geometric Functional Analysis and its Applications, Springer-Verlag, Berlin 1975.
[55] R. B. Holmes, B. Scranton, J.D. Ward, Approximation from the space of compact operators and other M-ideals, Duke Math. J. 42 (1975(, 259-269.
[56] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rat. Mech. Anal. 123 (1993), 51-74.
[57] J. A. Johnson, Banach spaces of Lipschitz functions and vector-valued Lipschitz functions, Trans. Amer. Math. Soc. 148 (1970), 147-171.
[58] J. A. Johnson, Lipschitz spaces, Pacific J. Math. 51 (1974), 177-186.
[59] J. A. Johnson, A note on Banach spaces of Lipschitz functions, Pacific J. Math. 58 (1975), 475-482.
[60] W. B. Johnson, J. Lindenstrauss and G. Schechtman, Extensions of Lipschitz maps into Banach spaces, Israel J. Math. 54 (1986), 129-138.
[61] V. M. Kadets, Lipschitz mappings on metric spaces, Matematika, Izvestija VUZOV (1985) no.1, 30-34, (in Russian).
[62] H. Kamowitz, S. Scheinberg, Some properties of endomorphisms of Lipschitz algebras, Studia Math. 96 (1990), 61-67.
[63] J. L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952), 323-326.
[64] M. Kirszbraun, Über die zusammenziehenden und Lipschitzschen Transformationen, Fund. Math. 22 (1934), 77108.
[65] S.V. Konyagin, On the level sets of Lipschitz functions, Tatra Mount. Math. Publ. 2 (1993), 51-59.
[66] A. Langenbach, Uber lokale eigenschaften von Lipschitz-Abbildungen, Math. Nachr. 194 (1998), 127-137.
[67] K. de Leeuw, Banach spaces of Lipschitz functions, Studia Math. 21 (1961), 55-66.
[68] R. Levi, M. D. Rice, The approximation of uniformly continuous mappings by Lipschitz and Hölder mappings, in General Topology and its Relations to Modern Analysis and Algebra V, Prague 1981, 455-461.
[69] J. Lindensstrauss, Non-linear projections in Banach spaces, Michigan Math. J. 11 (1964), 263-287.
[70] J. Lindenstrauss, Extension of compact operators, Memoirs Amer. Math. Soc. vol. 48 (1964).
[71] S. Mabizela, On bounded 2-linear functionals, Math. Japonica 35 (1990), 51-55.
[72] S. Mabizela, The relationship between Lipschitz extensions, best approximations, and continuous selections, Quaestiones Math. 14 (1991), 261-268.
[73] L. Marco, J. A. Murillo, Locally Lipschitz and convex functions, Mathematica 38 (1996), 121-131.
[74] J. Matoušek, Extension of Lipschitz mappings on metric trees, Comment. Math. Univ. Carol. 31 (1990), 99-101.
[75] J. Matoušek, Note on bi-Lipschitz embeddings into normed spaces, Comment. Math. Univ. Carol. 33 (1992), 51-55.
[76] J. Matoušek, On Lipschitz mappings onto square, in: The Mathematics of P. Erdös, vol. II, Springer-Verlag, Berlin 1997
[77] W. E. Mayer-Wolf, Isometries between Banach spaces of Lipschitz functions, Israel J. Math. 38 (1981), 58-74.
[78] J. A. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842.
[79] E. Mickle, On the extension of a transformation, Bull. Amer. Math. Soc. 55 (1949), 160-164.
[80] V. A. Milman, Extension of functions preserving the modulus of continuity, Mat. Zametki 61 (1997), 236-245 (in Russian).
[81] V. A. Milman, Lipschitz extensions of linearly bounded functions, Mat. Sbornik 189 (1998), 67-92 (in Russian).
[82] G. J. Minty, On the extension of Lipschitz-Hölder continuous, and monotone functions, Bull. Amer. Math. Soc. 76 (1970), 334-339.
[83] C. Mustăţa, On some Chebyshevian subspaces of the normed space of Lipschitz functions, Rev. Anal. Numer. Teor. Aprox. 2 (1973), 81-87 (in Romanian).
[84] C. Mustăţa, On the unicity of the extension of continuous p-seminorms, Rev. Anal. Numer. Teor. Aprox. Vol. 2 Fasc. 2 (1973), 173-177 (Romanian).
[85] C. Mustăţa, M-ideals in metric spaces, Babes-Bolyai University Reserch Seminaries, Preprint No. 7 (1988), 65-74.
[86] C. Mustăţa, On the best approximation in metric spaces, Rev. Anal. Numér. Théor. Approx. 4 (1975), 45-50.
[87] C. Mustăţa, Norm preserving extension of starshaped Lipschitz functions, Rev. Anal. Numér. Théor. Approx. 19 (1977), 183-187.
[88] C. Mustăţa, Best approximation and unique extension of Lipschitz functions, J. Approx. Theory 19 (1977), 222-230.
[89] C. Mustăţa, A characterization of Chebyshevian subspaces of Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥-type, Rev. Anal. Numér. Théor. Approx. 6 (1977), 51-56.
[90] C. Mustăţa, The extension of starshaped Lipschitz functions, Rev. Anal. Numér. Théor. Approx. 9 (1980), 93-99.
[91] C.Mustăţa, On the extension of Hölder functions, Babes-Bolyai University Reserch Seminaries, Seminar on Functional Analysis and Numerical Methods, Preprint No. 7 (1985), 71-86.
[92] C. Mustăţa, On the unicity of the extension of odd Lipschitz functions, Babes-Bolyai University, Seminar on Optimization Theory, Report No. 8 (1987), 75-80.
[93] C. Mustăţa, Extension of Hölder functions and some related problems of best approximation, Babes-Bolyai University, Faculty of Mathematics Research Seminaries, Preprint no. 7 (1991), 71-86.
[94] C. Mustăţa, Selections associated to McShane's extension theorem for Lipschitz functions, Rev. Anal. Numér. Théor. Approx. 21 (1992, 135-145.
[95] C. Mustăţa, On the metric projection and quotient mapping, Rev. Anal. Numér. Théor. Approx. 24 (1995, 191-199.
[96] L. Nachbin, On the Hahn-Banach theorem, Anais Acad. Brasil. Ci. 21 (1949), 151-154.
[97] L. Nachbin, A theorem of Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950), 28-46.
[98] L. Nachbin, Some problems in extending and lifting continuous linear transformations, in Proc. International Symp. Linear Spaces, Jerusalem 1960, Jerusalem Acad. Press and Pergamon Press, Jerusalem and London 1961, pp. 340350.
[99] Nguyen Van Khue, Nguyen To Xhu, Extending locally Lipschitz maps with values in an infinite dimensional nuclear Fréchet space, Bull. Acad. Polon. Sci. Ser. Math. 29 (1981), 609-616.
[100] Nguyen Van Khue, Nguyen To Xhu, Lipschitz extensions and Lipschitz retractions in metric spaces, Colloq. Math. 45 (1981), 245-250.
[101] E. Oja, On the uniqueness of the norm preserving extension of the linear functional in the Hahn-Banach theorem, Proc. Acad. Sci. Estonian SSR 33 (1984), 424-433.
[102] D. O'Keeffe, The tangent stars of a set, and extensions of functions in Lipschitz classes, Proc. Roy. Irish Acad. Sect. A, 97 (1997), 5-13.
[103] G. Pantelidis, Approximationstheorie für metrische lineare Räume, Math. Ann. 184 (1969), 30-48.
[104] Sung-Ho Park, Quotient mapping, Helly-extensions, Hahn-Banach extension, Tietze extension, Lipschitz extension and best approximation, J. Korean Math. Soc. 29 (1992), 239-250.
[105] J. Partanen, J. Vaisala, Extension of bi-Lipschitz maps on compact polyhedra, Math. Scand. 72 (1993), 235-264.
[106] R. R. Phelps, Uniqueness of Hahn-Banach extension and unique best approximation, Trans. Amer. Math. Soc. 95 (1960), 238-255.
[107] B. H. Pourciau, Analysis and optimization of Lipschitz continuous mappings, J. Optim. Theory Appl. 22 (1977), 311-351.
[108] B. H. Pourciau, Hadamard's theorem for locally Lipschitzian mappings, J. Math. Anal. Appl. 85 (1982), 279 − 285 279 − 285 279-285279-285279−285.
[109] B. H. Pourciau, Univalence and degree for Lipschitz continuous maps, Arch. Rat. Mech. Anal. 81 (1983), 289-299.
[110] B. H. Pourciau, Global properties of proper Lipschitzian maps, SIAM J. Math. Anal. 14 (1983), 796-799.
[111] K. Przeslawski, Lipschitz retracts, selections, and extensions, Michigan Math. J. 42 (1995), 555-571.
[112] K. Przeslawski, Lipschitz continuous selections. I. Linear selections, J. Convex Anal. 5 (1998), 249-267.
[113] S. Rolewicz, Metric Linear Spaces, PWN Warszawa 1972.
[114] S. Rolewicz, On Lipschitz projection-a geometrical approach, Ann. Univ. Maria Sklodowska-Curie 38 (1984), 135138.
[115] S. Rolewicz, On extremal points of the unit ball in Banach spaces of Lipschitz continuous functions, J. Austral. Math. Soc. Ser. A 41 (1986), 95-98.
[116] S. Rolewicz, Generalized Asplund inequalities on Lipschitz functions, Arch. Math. 61 (1993), 484-488.
[117] S. Rolewicz, On optimal observability of Lipschitz systems, in Selected Topics in Oper. Res. and Math. Economics, Lect. Notes in Economics and Math. Systems vol 226, pp. 151-158, Springer-Verlag, Berlin 1993.
[118] S. Rolewicz, On an extension of Mazur's theorem on Lipschitz functions, Arch. Math. 63 (1994), 535-540.
[119] S. Rolewicz, Duality and convex analysis in the absence of linear structure, Math. Japon. 44 (1996), 165-182.
[120] A. K. Roy, Extreme points and linear isometries of the Banach spaces of Lipschitz functions, Canad. J. Math. 20 (1968), 1150-1164.
[121] W. Ruess, Ein Dualkegel für p-konvexe topologische lineare Räume, Geselschaft für Mathematik und Datenverarbeitung Nr. 60 (1973).
[122] W. Rzymowski, Convex extension preserving Lipschitz constants, J. Math. Anal. Appl. 212 (1997), 30-37.
[123] I. Sawashima, Methods of duals in nonlinear analysis-Lipschitz duals of Banach spaces and some applications, Lect. Notes Econ. Math. Syst. vol. 419, Springer-Verlag 1975, pp.247-249.
[124] K. Schnatz, Approximationstheorie in metrischen Vektorräumen, Dissertation. Frankfurt am Main, 1985.
[125] K. Schnatz, Nonlinear duality and best approximation in metric linear spaces, J. Approx.Theory 49 (1987), 201-218.
[126] I. Schoenberg, On a theorem of Kirszbraun and Valentine, Amer. Math. Monthly 44 (1953), 620-622.
[127] S. O. Schönbeck, Extension of nonlinear contractions, Bull. Amer. Math. Soc. 72 (1966), 99-101.
[128] S. O. Schönbeck, On the extension of Lipschitz maps, Ark Math. 7 (1967-1969), 201-209.
[129] D. R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math. 13 (1963), 1387-1399.
[130] D. R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc., 111 (1964), 240-272.
[131] D. R. Shubert, A sequential method seeking the global maximum of function, SIAM J. Numer. Anal., 9 (1972), 379-388.
[132] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Ed. Academiei and Springer Verlag, Bucharest-Berlin 1970.
[133] I. Singer, Extension with larger norm and separation with double support in normed linear spaces, Bull. Austral. Math. Soc. 21 (1980), 93-106.
[134] R. Smarzewski, Extreme points of unit balls in Lipschitz functions spaces, Proc. Amer. Math. Soc. 125 (1997), 1391-1397.
[135] Ch. Stegall, Spaces of Lipschitz functions on Banach spaces, In Funct. Anal. (Essen 1991), 265-278. M. Dekker, New York 1994.
[136] V. Trenoguine, Analyse fonctionnelle, Edition Mir, Moscou 1985.
[137] S. Ustunel, Extension of Lipschitz functions on Wiener space, in Stochastic Anal. and Appl. (Powys 1995), 465-470, World Sci. 1996
[138] J. Vaisala, Bi-Lipschitz and quasi-symmetric extension properties, Ann. Acad. Sci. Fenn. SerI Math. 11 (1986), 239-274.
[139] J. Vaisala, Banach spaces and bi-Lipschitz maps, Studia Math. 103 (1992), 291-294.
[140] F. Valentine, A Lipschitz condition preserving extension of a vector function, Amer. J. Math. 67 (1945), 83-93.
[141] F. Valentine, On the extension of a vector function so as to preserve a Lipschitz condition, Bull. Amer. Math. Soc. 49 (1943), 100-108.
[142] F. Valentine, Contractions in non-euclidean spaces, Bull. Amer. Math. Soc. 50 (1944), 710-713.
[143] L. Waelbroeck, Closed ideals of Lipschitz functions, in Function Algebras (F. T. Birtel ed.), pp. 322-325, Scott Foresman 1966,
[144] N. Weaver, Lattices of Lipschitz functions, Pacific J. Math. 164 (1994), 179-193.
[145] N. Weaver, Isometries of noncompact Lipschitz spaces, Canad. Math. Bull. 38 (1995), 242-249.
[146] N. Weaver, Subalgebras of little Lipschitz algebras, Pacific J. Math. 173 (1996), 283-293.
[147] N. Weaver, Lipschitz algebras and derivations of von Neumann algebras, J. Func. Anal. 139 (1996), 261-300.
[148] J. H. Wells, L. R. Williams, Embeddings and Extensions in Analysis, Springer-Verlag, Berlin 1975.
[149] L. Williams, J. Wells, T. Hayden, On the extension of Lipschitz-Hölder maps on L p L p L^(p)L^{p}Lp spaces, Studia Math. 39 (1971), 29-38.
  1. Published in: Research on the Theory of Allure, Approximation, Convexity and Optimization, Elena Popoviciu - Editor, SRIMA Publishers, Cluj-Napoca 1999, pp. 3-21.
1999

Related Posts