On the extension of semi-Lipschitz functions on asymmetric normed spaces

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(original), Approximation Theory, asymmetric norm, paper

Abstract

Extension theorems for semi-Lipschitz functions and some proper- ties of these extensions useful in approximation problems are presented. As illustration, a such problem is considered.

Authors

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

Keywords

Spaces with asymmetric seminorm; semi-Lipschitz function; extension and approximation.

Paper coordinates

C. Mustăţa, On the extension of semi-Lipschitz functions on asymmetric normed spaces, Rev. Anal. Numer. Theor. Approx. 34 (2005) no. 2 , 139-150.

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https://ictp.acad.ro/jnaat/journal/article/view/2005-vol34-no2-art3/Mustata-pdf

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Journal

Revue d’Analyse Numer. Theor. Approx.

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Publishing House of the Romanian Academy

DOI

https://ictp.acad.ro/jnaat/journal/article/view/2005-vol34-no2-art3

Print ISSN

2501-059X

Online ISSN

2457-6794

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[5] Garcia-Raffi, L.M.,Romaguera, S. and Sanchez-Perez, E.A.,The dual space ofan asymmetric normed linear space, Quaest. Math.,26, no. 1, pp. 83–96, 2003.
[6] McShane, E.J.,Extension of range of functions, Bull. Amer. Math. Soc.,40, pp. 837–842, 1934.
[7] Mustata, C.,On a chebyshevian subspace of normed linear space of Lipschitz func-tions, Rev. Anal. Numer. Teoria Aproximat ̧iei,2, pp. 81–87, 1973 (in Romanian).
[8] Mustata, C.,Best approximation and unique extension of Lipschitz functions, J. Ap-prox. Theory,19, no. 3, pp. 222–230, 1977.
[9] Mustata, C.,Extension of Holder functions and some related problems of best ap-proximation, “Babe ̧s-Bolyai” University, Faculty of Mathematics,Research Seminar onMathematical Analysis, no. 7, pp. 71–86, 1991.
[10] Mustata, C.,Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal.Numer. Theor. Approx.,30, no. 1, pp. 61–67, 2001.
[11] Mustata, C.,The approximation of the global maximum of a semi-Lipschitz function(submitted).
[12] Leonardi, S., Passarelli di Napoli, A.and Carlo Sbordone,On Fichera’s ex-istence principle in functional analysis and mathematical Physiscs, Papers of the 2-ndInterantional Symposium dedicated to memory of Prof. Gaetano Fichera (1922–1996).Roma: Dipartimento di Matematica Univ. di Roma (ISBN 88-7999-264-X), pp. 221–2342000, Ricci, PaoloEmilio (Ed.)
[13] Romaguerra, S. and Sanchis, M.,Semi-Lipschitz functions and best approximationin quasi-metric spaces, J. Approx. Theory,103, pp. 292–301, 2000.

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ON THE EXTENSION OF SEMI-LIPSCHITZ FUNCTIONS ON ASYMMETRIC NORMED SPACES COSTIC ˘ A MUST ˘ AT¸A * Abstract. Extension theorems for semi-Lipschitz functions and some proper- ties of these extensions useful in approximation problems are presented. As illustration, a such problem is considered. MSC 2000. 46A22, 26A16,41A50. Keywords. Spaces with asymmetric seminorm, semi-Lipschitz function, exten- sion and approximation. 1. PRELIMINARIES Let X be a real linear space. A function p : X → [0, ∞) is called an asymmetric seminorm [2] if the following conditions hold for all x, y ∈ X : p (x) ≥ 0, (AN1) p(tx)= tp (x) , t ≥ 0, (AN2) p (x + y) ≤ p (x)+ p (y) . (AN3) The function p : X → [0, ∞) deﬁned by p (x)= p (-x) ,x ∈ X, is also an asymmetric seminorm on X, called the conjugate seminorm to p. The func- tional p s (x) = max{p (x) ,p (-x)}, x ∈ X, is a seminorm on X. If p s is a norm on X, then p is called an asymmetric norm. The term “asymmetric” is motivated by the fact that it is possible that p (x) = p (-x) for some x ∈ X. A pair (X, p) where X is a real linear space and p an asymmetric seminorm on X, is called an asymmetric seminormed space, respectively asymmetric normed space if p is an asymmetric norm on X. Some properties of such spaces are given in [3], [4], and the references therein. An example is the following: let R be the set of the real numbers and u : R → [0, ∞), u (a) = max{a, 0},a ∈ R. Then the function u is an asymmetric norm on R. The conjugate u : R → [0, ∞), u (a)= u (-a) ,a ∈ R, is another * “T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro.

asymmetric norm on R, because the function u s (a) = max {u(a),u(-a)} = |a| , a ∈ R, is a norm on R (see [4], [5]). If (X, p) is an asymmetric seminormed space, one considers the following topologies on X : (1) The topology τ p generated by the families of forward open balls B + (x, ε)= {z ∈ X : p (z - x) <ε},x ∈ X, ε > 0; (2) The topology τ p generated by the families of backward open balls B - (x, ε)= {z ∈ X : p (x - z ) <ε},x ∈ X, ε > 0; (3) The topology τ generated by the open balls B (x, ε)= {z ∈ X : p s (x - z ) <ε} = B + (x, ε) ∩ B - (x, ε) ,x ∈ X, ε > 0. For details on these topologies, see [1] and [3]. 2. THE CONES OF SEMI-LIPSCHITZ FUNCTIONS Let (X, p) be an asymmetric seminormed space and Y a subset of X. A function f : Y → R is called p-semi-Lipschitz if there exists a constant K ≥ 0 such that (1) f (u) - f (v) ≤ Kp (u - v) , for all u, v ∈ X, (see [13]). Observe that a p-semi-Lipschitz function is upper semicontinuous. The set of all p-semi-Lipschitz functions on Y is denoted by symbol p-SLipY , i.e. (2) p-SLipY := {f : Y → R,f is p-semi-Lipschitz}. If f ∈p-SLipY , then the nonnegative number (3) ||f | p = sup  (f (u)-f (v))∨0 p(u-v) : u, v ∈ Y,p (u - v) > 0  is the smallest p-semi-Lipschitz constant for F, i.e. the following inequality holds: f (u) - f (v) ≤ ||f | p p (u - v) , for all u, v ∈ Y (see [10]). The set p-SLipY is closed with respect to pointwise addition of functions, and with respect to multiplication of a function by nonnegative scalar, i.e. p-SLipY is a cone. The functional ||·| p : p-SLipY → [0, ∞) deﬁned by (3) veriﬁes the properties (AN 1)–(AN 3) of an asymmetric seminorm. Analogously, a function f : Y → R is called p-semi-Lipschitz if there exists Q ≥ 0 such that (4) f (u) - f (v) ≤ Q ¯ p (u - v)= Qp (v - u) , for all u, v ∈ Y. The smallest constant Q in (4) is denoted by ||f | p and the set of all p- semi-Lipschitz functions on Y is denoted by symbol ¯ p-SLipY . The functional

|| · | p :¯ p-SLipY → [0, ∞) deﬁned by an expression analogue with (3) is an asymmetric norm on the cone ¯ p-SLipY . The function f : Y → R is called p s -Lipschitz if there exits M ≥ 0 such that (5) |f (u) - f (v) |≤ Mp s (u - v) , for all u, v ∈ Y. The set of all p s -Lipschitz functions on Y is denoted by symbol p s -SLipY . With respect to pointwise addition of functions and multiplication of functions by real numbers, the set p s -SLipY is a linear space. The functional · :p s -SLipY → [0, ∞), (6) f  = sup  |f (u)-f (v)| p s (u-v) : u, v ∈ Y,p s (u - v) > 0  is a seminorm on linear space p s -SLipY . Proposition 1. Let (X, p) be an asymmetric seminormed space and Y a subset of X . Then a) The sets p-SLipY and ¯ p-SLipY are convex cones in the space p s -SLipY ; b) The function f is in p-SLipY if and only if -f is in ¯ p-SLipY . For every f ∈p-SLipY , the following equality holds: ||f | p = || - f | p = f . Proof. a) Let f in p-SLipY . Then f (u) - f (v) ≤ ||f | p p (u - v) ≤ ||f | p p s (u - v) , for all u, v ∈ Y. Changing the place of u and v, we obtain f (v) - f (u) ≤ ||f | p p (v - u)= ||f | p p (u - v) ≤ ||f | p p s (u - v) , for all u, v ∈ Y. It follows that |f (u) - f (v)| ≤ ||f | p p s (u - v) , for all u, v ∈ Y. Consequently f ∈p s -SLipY (and f  = ||f | p ). If f ∈ ¯ p-SLipY , one obtains |f (u) - f (v)| ≤ ||f | p p s (u - v) , for all u, v ∈ Y, i.e. f ∈ ¯ p-SLipY , (it follows f  = ||f | p ). b) Let f ∈p-SLipY . By f (u) - f (v) ≤ ||f | p p (u - v), for all u, v ∈ Y, it follows (-f )(v) - (-f )(u) ≤ ||f | p p (u - v)= ||f | p p (v - u) , and then -f ∈ ¯ p-SLipY . Moreover, it follows that ||- f | p ≤ ||f | p . Analogously, f ∈ ¯ p-SLipY implies (-f )(v) - (-f )(u) ≤ ||f | p p (u - v)= ||f | p p (v - u) , for all u, v ∈ Y. Consequently -f ∈p-SLipY and || - f | p ≤ ||f | p .

Observe that for f ∈p-SLipY , ||f | p = || - (-f ) | p ≤ || - f | p , and for f ∈ ¯ p-SLipY , one obtains ||f | p = || - (-f ) | p ≤ || - f | p . Finally, it follows ||f | p = || - f | p , ∀f ∈ p-SLipY, and ||f | p = || - f | p , ∀f ∈ ¯ p-SLipY.  Remarks. 1) By Proposition 1, a) it follows that for f ∈p-SLipY , (g ∈ ¯ p-SLipY ) one obtains ||f | p = f  , (respectively ||g| p = g) and con- sequently f ∈ p-SLipY ∩ ¯ p-SLipY ⇒ ||f | p = ||f | p = f  . 2) Let y 0 be a ﬁxed element in Y. Denote p s -SLip 0 Y := {f ∈ p s -LipY,f (y 0 )=0} , p-SLip 0 Y := {f ∈ p-SLipY,f (y 0 )=0} , ¯ p-SLip 0 Y := {f ∈ ¯ p-SLipY,f (y 0 )=0} . If f ∈p s -SLip 0 Y , then f  = 0 implies f ≡ 0. It follows that f ∈ p-SLip 0 Y ∩ ¯ p-SLip 0 Y implies f ≡ 0 and ||f | p , ||f | p are asymmetric norms on p-SLip 0 Y and ¯ p-SLip 0 Y respectively. 3) Let Y be a subspace of asymmetric normed space (X, p) and ϕ : Y → R a linear functional. If (R,u) is the asymmetric normed space with asymmetric norm u (a) = max{a, 0},a ∈ R and τ u is the topology associated to u, then the functional ϕ : Y → R is called (p, u)-continuous if it is continuous in the topologies τ p and τ u .  The linear functional ϕ 🙁Y,p) → (R,u) is called p-bounded if there exists L ≥ 0 such that ϕ (y) ≤ Lp (y) , for every y ∈ Y. The functional ϕ 🙁Y,p) → (R,u) is (p, u)-continuous if and only if ϕ is p-bounded, (see [4]). Every p-bounded linear functional on Y is p-semi-Lipschitz on Y. Denote by Y b p the set of all p-bounded functionals on Y. Then Y b p ⊂p-SLip 0 Y (here y 0 = 0), and for ϕ ∈ Y b p , (7) ||ϕ| p = sup y=0 ϕ (y) p (y) = sup {ϕ (y): y ∈ Y,p (y) ≤ 1} . The functional || · | p : Y b p → [0, ∞) deﬁned by (7) is an asymmetric norm, and the pair  Y b p , || · | p  is called asymmetric dual cone of (Y,p) [5].

3. EXTENSION RESULTS Let Y be a subset of asymmetric normed space (X, p) . The function f : Y → R is called bounded on Y if there exists the numbers m, M ∈ R such that m ≤ f (y) ≤ M, for every y ∈ Y. Proposition 2. Let (X, p) be an asymmetric normed space, Y a subset of X and f : Y → R a bounded function. Let also K ≥ 0 be arbitrary, but ﬁxed. Then the functions F p (f ) ,G p (f ): X → R deﬁned by the formulas F p (f )(x) = inf y∈Y {f (y)+ Kp (x - y)} , x ∈ X, G p (f )(x) = sup y∈Y {f (y) - Kp (y - x)} , x ∈ X, satisfy the following relations: F p (f )(y) ≤ f (y) ≤ G p (f )(y) , ∀y ∈ Y, (a) F p (f )(x 1 ) - F p (f )(x 2 ) ≤ Kp (x 1 - x 2 ) , ∀x 1 ,x 2 ∈ X, (b) G p (f )(x 1 ) - G p (f )(x 2 ) ≤ Kp (x 1 - x 2 ) , ∀x 1 ,x 2 ∈ X. (c) Proof. Let f : Y → R and m, M ∈ R such that m ≤ f (y) ≤ M for every y ∈ Y. It follows m ≤ f (y)+ Kp (x - y) , ∀y ∈ Y, ∀x ∈ X, and M ≥ f (y) - Kp (y - x) , ∀y ∈ Y, ∀x ∈ X. Consequently, the set {f (y)+ Kp (x - y) | y ∈ Y } is bounded from below for every x ∈ X and the set {f (y) - Kp (y - x) |y ∈ Y } is bounded from above for every x ∈ X. Then the functions F p (f ) and G p (f ) are well deﬁned on X. a) Let y 0 ∈ Y be ﬁxed. For every x ∈ X, inf y∈Y {f (y)+ Kp (x - y)}≤ f (y 0 )+ Kp (x - y 0 ) , and for x = y 0 it follows that F p (f )(y 0 ) ≤ f (y 0 ) . Analogously, sup y∈Y {f (y) - Kp (y - x)}≥ f (y 0 ) - Kp (y 0 - x) , and for x = y 0 , we obtain f (y 0 ) ≤ G p (f )(y 0 ) . Because y 0 is arbitrary in Y it follows that F p (f )(y) ≤ f (y) ≤ G p (f )(y), for every y ∈ Y.

b) Let x 1 ,x 2 ∈ X. Because for every y ∈ Y we have 0 ≤ p (x 1 - y) = p (x 1 - x 2 + x 2 - y) ≤ p (x 1 - x 2 )+ p (x 2 - y) , it follows the inequality f (y)+ Kp (x 1 - y) ≤ f (y)+ Kp (x 1 - x 2 )+ Kp (x 2 - y) , and taking the inﬁmum with respect to y ∈ Y one obtain F p (f )(x 1 ) ≤ F p (f )(x 2 )+ Kp (x 1 - x 2 ) . Then F p (f )(x 1 ) - F p (f )(x 2 ) ≤ Kp (x 1 - x 2 ) , for every x 1 ,x 2 ∈ X. c) Using the inequalities 0 ≤ p (y - x 2 )= p (y - x 1 + x 1 - x 2 ) ≤ ≤ p (y - x 1 )+ p (x 1 - x 2 ) , one obtains f (y) - Kp (y - x 2 ) ≥ f (y) - Kp (y - x 1 ) - Kp (x 1 - x 2 ) and taking the supremum with respect to y ∈ Y we obtain G p (f )(x 1 ) - G p (f )(x 2 ) ≤ Kp (x 1 - x 2 ) .  A subset Y of an asymmetric normed space (X, p) is called p-bounded,( p- bounded ) if there exists M ≥ 0(Q ≥ 0) such that p (u - v) ≤ M ( p (u - v) ≤ Q) , for all u, v ∈ Y. The set Y is called (p, p)-bounded if it is both p-bounded and p-bounded. Proposition 3. If Y is a (p, p)-bounded set of an asymmetric normed space (X, p) and f ∈p-SLipY , then f is bounded. Proof. Let y 0 ∈ Y be ﬁxed. For every y ∈ Y, f (y) -f (y 0 ) ≤ ||f | p p (y - y 0 ) , implies f (y) ≤ f (y 0 )+ ||f | p p (y - y 0 ) . Analogously, f (y 0 ) - f (y) ≤ ||f | p p (y 0 - y)= ||f | p ¯ p (y - y 0 ) , for every y ∈ Y. Then f (y 0 ) - ||f | p p (y - y 0 ) ≤ f (y) ≤ f (y 0 )+ ||f | p p (y - y 0 ) , and because Y is (p, p) -bounded, it follows that f is bounded.  Remark 1. By Proposition 1, it follows that every g ∈ ¯ p-SLipY is bounded, if Y is (p, p)-bounded. For the symmetric case of Proposition 2, see [12]. 

Proposition 4. Let (X, p) be an asymmetric normed space, Y a (p, p)- bounded subset of X and f ∈p-SLipY . Let ||f | p the smallest semi-Lipschitz constant of f. Then the functions F p (f ) ,G p (f ): X → R deﬁned as in Propo- sition 2 with K = ||f | p satisfy the following properties: a) G p (f )(x) ≤ F p (f )(x) ,x ∈ X, b) G p (f )(y)= f (y)= F p (f )(y) ,y ∈ Y, c) ||G p (f ) | p = ||f | p = ||G p (f ) | p , d) If g : X → R and g ∈p-SLipX veriﬁes g| Y = f and ||f | p = ||g| p , then G p (f )(x) ≤ g (x) ≤ F p (f )(x) , for every x ∈ X. Proof. a) For Every u, v ∈ Y and x ∈ X, the following inequalities are fulﬁlled: f (u) - f (v) ≤ ||f | p p (u - v) ≤ ||f | p p (u - x)+ ||f | p p (x - v) . Then f (u) - ||f | p p (u - x) ≤ f (v)+ p (x - v) . Taking the inﬁmum with respect to v ∈ Y and then with respect to u ∈ Y, we obtain G p (f )(x) ≤ F p (f )(x) , for every x ∈ X. b) It follows from Proposition 2 a), Proposition 3 and the previous inequality of a). c) The following inequalities are obvious: ||G p (f ) | p ≥ ||f | p and ||F p (f ) | p ≥ ||f | p . Let now x 1 ,x 2 ∈ X and ε> 0. Selecting y ∈ Y such that F p (f )(x 1 ) ≥ f (y)+ ||f | p p (x 1 - y) - ε, one obtains: F p (f )(x 2 ) - F p (f )(x 1 ) ≤ ≤ f (y)+ ||f | p p (x 2 - y) - (f (y)+ ||f | p p (x 1 - y) - ε) = ||f | p [p (x 2 - y) - p (x 1 - y)] + ε ≤ ||f | p p (x 2 - x 1 ) (since p (x 2 - y) - p (x 1 - y) ≤ p (x 2 - x 1 ) ⇐⇒ p (x 2 - y)= p(x 2 - x 1 + x 1 - y) ≤ p (x 2 - x 1 )+ p (x 1 - y)). The number ε> 0 being arbitrarily chosen, it follows F p (f )(x 2 ) - F p (f )(x 1 ) ≤ ||f | p p (x 2 - x 1 ) , for all x 1 ,x 2 ∈ X, and then ||F p (f ) | p ≤ ||f | p .

Analogously, we obtain ||G p (f ) | p ≤ ||f | p , and consequently ||G p (f ) | p = ||f | p = ||F p (f ) | p . d) Let g ∈p-SLipX such that g| Y = f , and ||g| p = ||f | p . For every y ∈ Y and x ∈ X we have f (y) - g (x)= g (y) - g (x) ≤ ||g| p p (y - x) , and then f (y) - ||g| p p (y - x) ≤ g (x) . Taking the supremum with respect to y ∈ Y we obtain G p (f )(x) ≤ g (x) , for every x ∈ X. Analogously, g (x) - f (y)= g (x) - g (y) ≤ ||g| p p (x - y)= ||f | p p (x - y) , implies g (x) ≤ f (y)+ ||f | p p (x - y) and taking the inﬁmum with respect to y ∈ Y, it follows g (x) ≤ F p (f )(x) , for every x ∈ X.  Remarks. 1) In Proposition 4 the condition “Y is a (p, p)-bounded set ” is not necessary. Indeed, for a nonvoid set Y and f ∈p-SLipY , the functions F p (f ) and G p (f ) are well deﬁned. For this, let y 0 ∈ Y and x ∈ X. For every y ∈ Y, f (y)+ ||f | p p (x - y)= f (y 0 )+ ||f | p p (x - y) - (f (y 0 ) - f (y)) ≥ f (y 0 )+ ||f | p [p (x - y) - p (y 0 - y)] = f (y 0 ) - ||f | p [p (y 0 - y) - p (x - y)]. But p (y 0 - y) - p (x - y) ≤ p (y 0 - x)= p (x - y 0 ) , and then the set {f (y)+ ||f | p p (x - y): y ∈ Y } is bounded below and there exists F p (f )(x) = inf y∈Y {f (y)+ ||f | p p (x - y)}, for every x ∈ X. Analogously, the function G p (f )(x) = sup y∈Y {f (y) - ||f | p p (y - x)} is well deﬁned for every x ∈ X. 2) A proposition similar to Proposition 4 is valid for the functions of cone ¯ p-SLipY . 3) For f ∈p-SLipY (f ∈ ¯ p-SLipY ) let E p (f,Y )= {g ∈ p-SLipX : g| Y = f and ||g| p = ||f | p },

and respectively E p (f,Y )= {h ∈ ¯ p-SLipX : h| Y = f and ||h| p = ||f | p }, the sets of extensions preserving the asymmetric seminorms ||f | p (respectively ||f | p ). We have the following inclusions: E p (f,Y ) ⊂ S + (0, ||f | p )= {g ∈ p-SLipX : g| p = ||f | p }, E p (f,Y ) ⊂ S - (0, ||f | p )= {h ∈ ¯ p-SLipX : ||h| p = ||f | p }. 4) By Theorem of McShane [6], for every f ∈p s -SLipY there exists F ∈p s -SLipX , such that F | Y = f and ||F || = ||f ||, where ||f || is deﬁned by formulas (6) and ||F || analogously. Denote by E p s (f,Y )= {F ∈ p s -SLipX : F | Y = f and ||F || = ||f ||}, the set of all extensions of f ∈p s -SLipY preserving the Lipschitz constant ||f ||. By Proposition 1, if f ∈p-SLipY , then f ∈p s -SLipY and because ||f | p ≤ ||f ||, it follows that F p (f )(x) ≤ F (f )(x) , x ∈ X, where F (f )(x) = inf y∈Y {f (y)+ ||f || p s (x - y)}. If f ∈ ¯ p-SLipY , then -f ∈p-SLipY and F p (-f )(x) ≤ F (-f )(x) ,x ∈ X. But F p (-f )(x) = inf y∈Y {-f (y)+ || - f | p p (x - y)} = - sup y∈Y {f (y) - || - f | p p (x - y)} = - sup y∈Y {f (y) - || - f | p p (x - y)} = -G p (f )(x) , for every x ∈ X. Analogously, if f ∈p-SLipY , then -f ∈ ¯ p-SLipY , and F p (-f )(x)= -G p (f )(x) , ∀x ∈ X. If G (f )(x) = sup y∈Y {f (y) - || - f || p s (x - y)}, then F (-f )(x)= -G (f )(x) , x ∈ X, and then -G p (f )(x) ≤-G (f )(x) , and G (f )(x) ≤ G p (f )(x) , x ∈ X.

It follows that, if f ∈p-SLipY , then G (f ) ≤ G p (f ) ≤ F p (f ) ≤ F (f ) on X, and for f ∈ ¯ p-SLipY , G (f ) ≤ G p (f ) ≤ F p (f ) ≤ F (f ) on X. 5) Let Y be a subspace of asymmetric normed space (X, p) and let ϕ 0 a bounded linear functional  ϕ 0 ∈ Y b p  . By Proposition 3.1 [4] there exists ϕ ∈ X b p such that ϕ| Y = ϕ 0 and ||ϕ| p = ||ϕ 0 | p . Let E p (ϕ 0 )= {ϕ ∈ X b p : ϕ| Y = ϕ 0 and ||ϕ| p = ||ϕ 0 | p } be the set of all extensions of ϕ 0 which preserves the asymmetric norm ||ϕ 0 | p . In this case Y b p ⊂p-SLip 0 Y and for every ϕ ∈ E p (ϕ 0 ) the following inequal- ities hold: G p (ϕ)(x) ≤ ϕ (x) ≤ F p (ϕ)(x) , x ∈ X. If ϕ 0 ∈ Y b p but ϕ 0 / ∈ Y b p , then ϕ 0  B + (0,r)  =(-∞,r||ϕ 0 | p ), r> 0. If ψ 0 ∈ Y b p but ψ 0 / ∈ Y b p, , then ψ 0 ( B - (0,r) ) =(-∞,r||ψ 0 | p ), r> 0. If ϕ ∈ Y b p ∩ Y b p, , then ϕ  B + (0,r)  =(-r||ϕ| p ,r||ϕ| p ), r> 0, ϕ ( B - (0,r) ) =(-r||ϕ| p ,r||ϕ| p ), r> 0.  Example 1. [11] (see also [9]) Let (R,u) be the asymmetric normed space with u (a) = max {a, 0},a ∈ R. Then u (a) = max{-a, 0} and u s (a)= |a|,a ∈ R. Let Y =[-1, 2] and f : Y → R , f (y)=4y - y 2 . Because f (y 1 ) - f (y 2 ) ≤ 6 max{y 1 - y 2 , 0} for all y 1 ,y 2 ∈ [-1, 2], it follows that f ∈ ¯ u-SLipY and ||f | u = 6. Then the functions F u (f )(x)=    -5, x ∈ (-∞, -1), 4x - x 2 , x ∈ [-1, 2], 6x - 8, x ∈ (2, ∞) and G u (f )(x)=    6x +1, x ∈ (-∞, -1), 4x - x 2 , x ∈ [-1, 2], 4, x ∈ (2, ∞) are the maximal and minimal extensions of f on (R,u) with the asymmetric norms ||F u (f ) | u = ||G u (f ) | u = ||f || u =6.

If one considers the normed space (R,u s ) , then the functions F (f )(x)= inf y∈[-1,2] {f (y)+6|x - y|}, x ∈ R, G (f )(x)= sup y∈[-1,2] {f (y)+6|x - y|}, x ∈ R are Lipschtiz extensions for f and G (f )(x) ≤ G u (f )(x) ≤ F u (f )(x) ≤ F (f )(x) , x ∈ R. 4. APPLICATION Let (X, p) be an asymetric normed space, and p s -SLip 0 X the normed space of all p s -Lipschitz real functions on X, vanishing at 0 ∈ X, with the norm deﬁned by (6). If Y is a subset of X with 0 ∈ Y let also the normed space p s -SLip 0 Y , with the norm · . By the McShane theorem [6] for every f ∈p s -SLip 0 Y there exists at least one function F ∈p s -SLip 0 X such that F | Y = f and F  = f  . Let also Y ⊥ = {G ∈ p s -SLip 0 X : G| Y =0} , the annihilator of Y with respect to p s -SLip 0 X . Consider the following best approximation problem: for F ∈p s -SLip 0 X , ﬁnd an element G 0 ∈ Y ⊥ such that F - G 0  = d(F,Y ⊥ ) = inf  F - G : G ∈ Y ⊥  . An element G 0 ∈ Y ⊥ such that the above inﬁmum is attained is called best approximation element for F in Y ⊥ . If every F ∈p s -SLip 0 X has at least a best approximation element, then Y ⊥ is called proximinal. The following result appears in [7] (see also [8]). Proposition 5. In the above notations, the following properties hold: (a) d(F,Y ⊥ )= F | Y  , ∀F ∈ p s -SLip 0 X ; (b) the set of all best approximation elements of F in Y ⊥ is F -E p s (F | Y ,Y ), where E p s (F | Y ,Y )=  H ∈ p s -SLip 0 X : H | Y = F | Y and H  = F | Y   . Proof. (a) Let F ∈p s -SLip 0 Y . Then for every G ∈ Y ⊥ , F | Y  = F | Y - G| Y ≤F - G, and taking the inﬁmum with respect to G ∈ Y ⊥ one obtains F | Y ≤ d(F,Y ⊥ ). For F | Y , by the McShane theorem [6], there exists H ∈ E p s (F | Y ,Y ) such that F | Y = H | Y and H  = F | Y  . Then F | Y  = F - (F - H )≥ d(F,Y ⊥ ).

(b) Obviously, for every best approximation element G 0 of F in Y ⊥ , (F - G 0 )| Y = F | Y and F - G 0  = F | Y  . It follows that F - G 0 ∈ F - E p s (F | Y ,Y ), and so G 0 ∈ F - E p s (F | Y ,Y ). Let now G 0 ∈ F - E p s (F | Y ,Y ). Then there exists H ∈ E p s (F | Y ,Y ) such that G 0 = F - H. But then F - G 0  = H  = F | Y  = d(F,Y ⊥ ).  Remark 2. If F ∈ ¯ p-SLip 0 X then F  = F | p and F | Y ∈p-SLip 0 Y , F | Y  = F | Y | p = - F | Y | ¯ p . It then follows d(F,Y ⊥ )= d(-F,Y ⊥ )= F | Y | p , ∀F ∈ p-SLip 0 X.  REFERENCES [1] Mennucci, Andrea C.G., On asymmetric distances, preprint, sept. 21, 2004 (www.scirus.com). [2] Borodin, P.A., The Banach-Mazur theorem for spaces with an asymmetric norm and its applications in convex analysis, Mat. Zametki, 69, no. 3, pp. 193–217, 2001. [3] Cobzas ¸, S., Separation of convex sets and best approximation in spaces with asymmetric norm, Quaest. Math., 27, no. 3, pp. 275–296, 2004. [4] Cobzas, S. and Must˘ at¸a, C., Extension of bouned linear functionals and best approx- imation in spaces with asymmetric norm, Rev. Anal. Numer. Theor. Approx., 33, no. 1, pp. 39–50, 2004. [5] Garcia-Raffi, L.M., Romaguera, S. and Sanchez-Perez, E.A., The dual space of an asymmetric normed linear space, Quaest. Math., 26, no. 1, pp. 83–96, 2003. [6] McShane, E.J., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837– 842, 1934. [7] Must˘ at¸a, C., On a chebyshevian subspace of normed linear space of Lipschitz func- tions, Rev. Anal. Numer. Teoria Aproximat ¸iei, 2, pp. 81–87, 1973 (in Romanian). [8] Must˘ at¸a, C., Best approximation and unique extension of Lipschitz functions, J. Ap- prox. Theory, 19, no. 3, pp. 222–230, 1977. [9] Must˘ at¸a, C., Extension of H¨ older functions and some related problems of best ap- proximation, “Babe¸s-Bolyai” University, Faculty of Mathematics, Research Seminar on Mathematical Analysis, no. 7, pp. 71–86, 1991. [10] Must˘ at¸a, C., Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numer. Theor. Approx., 30, no. 1, pp. 61–67, 2001. [11] Must˘ at¸a, C., The approximation of the global maximum of a semi-Lipschitz function (submitted). [12] Leonardi, S., Passarelli di Napoli, A. and Carlo Sbordone, On Fichera’s ex- istence principle in functional analysis and mathematical Physiscs, Papers of the 2-nd Interantional Symposium dedicated to memory of Prof. Gaetano Fichera (1922–1996). Roma: Dipartimento di Matematica Univ. di Roma (ISBN 88-7999-264-X), pp. 221–234 2000, Ricci, PaoloEmilio (Ed.) [13] Romaguerra, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103, pp. 292–301, 2000. Received by the editors: March 28, 2005.