Abstract
The extremal elements of the unit balls of Banach spaces play an important role in the study of the geometry of the space as well in various applications. For Banach spaces of Lipschitz real functions the extremal elements of the unit ball are investigates in numerous papers (S. Cobzas 1989, J. D. Farmer 1994, N. V. Rao and A. C. Roy 1970, Roy 1968 and in the references therein). In this note we shall present a procedure to obtain extremal elements of the unit ball of the quasi normed semilinear space of real-valued semi-Lipschitz functions defined on a quasi-metric space.
Authors
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
semi metric spaces; semi Lipschitz real functions.
Paper coordinates
C. Mustăţa, On the extremal semi-Lipschitz function, Rev. Anal. Numer. Theor. Approx. 31 (2002) no. 1, 103-108, https://ictp.acad.ro/jnaat/journal/article/view/2002-vol31-no1-art11
About this paper
Journal
Revue d’Analyse Numer.Theor. Approx.
Publisher Name
Publishing Romanian Academy
Print ISSN
2457-6794
Online ISSN
2501-059X
google scholar link
[1] Cobzas, S. and Mustata, C., Norm preserving extension of convex Lipschitz functions, J. Approx. Theory,24, pp. 555–564, 1978.
[2] Cobzas, S., Extreme points in Banach spaces of Lipschitz functions, Mathematica, 31(54), pp. 25–33, 1989.
[3] Farmer, J. D., Extreme points of the unit ball of the space of Lipschitz functions, Proc. Amer. Math. Soc., 121, no. 3, pp. 807–813, 1994.
[4] Krein, M. G. and A. A. Nudel’man, The Markov Moment Problem and Extremum Problems, Nauka, Moscow, 1973, (in Russian).
[5] Koppermann, R. D., All topologies come from generalized metrics, Amer. Math. Monthly, 95, pp. 89–97, 1988.
[6] McShane, E. J., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837–842, 1934.
[7] Mustata, C., Best approximation and unique extension of Lipschitz functions, J. Approx. Theory, 19, pp. 222–230, 1977.
[8] Mustata, C., Uniquenness of the extension of semi-Lipschitz functions on quasi-metric spaces, Bull. S ̧t. Univ. Baia Mare, Mat.-Inf., XVI, no. 2, pp. 207–212, 2000.
[9] Mustata, C., Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numer. Theor. Approx., 2001 (to appear).
[10] Phelps, R. R., Uniqueness of Hahn-Banach extension and unique best approximation, Trans. Amer. Math. Soc., 95, pp. 238–255, 1960.
[11] Rao, N. V. and Roy, A. K., Extreme Lipschitz functions, Math. Ann., 189, pp. 26–46, 1970.
[12] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric space, J. Approx. Theory, 103, pp. 293-301, 2000.
[13] Roy, A. K., Extreme points and linear isometries of Banach spaces of Lipschitz functions, Canad. J. Math., 20, pp. 1150–1164, 1968.
[14] Wells, J. H. and Williams, L. R., Embedings and Extensions in Analysis, Springer-Verlag, Berlin, 1975.
[2] Cobzas, S., Extreme points in Banach spaces of Lipschitz functions, Mathematica, 31(54), pp. 25–33, 1989.
[3] Farmer, J. D., Extreme points of the unit ball of the space of Lipschitz functions, Proc. Amer. Math. Soc., 121, no. 3, pp. 807–813, 1994.
[4] Krein, M. G. and A. A. Nudel’man, The Markov Moment Problem and Extremum Problems, Nauka, Moscow, 1973, (in Russian).
[5] Koppermann, R. D., All topologies come from generalized metrics, Amer. Math. Monthly, 95, pp. 89–97, 1988.
[6] McShane, E. J., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837–842, 1934.
[7] Mustata, C., Best approximation and unique extension of Lipschitz functions, J. Approx. Theory, 19, pp. 222–230, 1977.
[8] Mustata, C., Uniquenness of the extension of semi-Lipschitz functions on quasi-metric spaces, Bull. S ̧t. Univ. Baia Mare, Mat.-Inf., XVI, no. 2, pp. 207–212, 2000.
[9] Mustata, C., Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numer. Theor. Approx., 2001 (to appear).
[10] Phelps, R. R., Uniqueness of Hahn-Banach extension and unique best approximation, Trans. Amer. Math. Soc., 95, pp. 238–255, 1960.
[11] Rao, N. V. and Roy, A. K., Extreme Lipschitz functions, Math. Ann., 189, pp. 26–46, 1970.
[12] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric space, J. Approx. Theory, 103, pp. 293-301, 2000.
[13] Roy, A. K., Extreme points and linear isometries of Banach spaces of Lipschitz functions, Canad. J. Math., 20, pp. 1150–1164, 1968.
[14] Wells, J. H. and Williams, L. R., Embedings and Extensions in Analysis, Springer-Verlag, Berlin, 1975.
Paper (preprint) in HTML form
ON THE EXTREMAL SEMI-LIPSCHITZ FUNCTIONS
COSTICĂ MUSTĂŢA*
Abstract
The extremal elements of the unit balls of Banach spaces play an important role in the study of the geometry of the space as well in various applications. For Banach spaces of Lipschitz real functions the extremal elements of the unit ball are investigates in numerous papers (S. Cobzas 1989, J. D. Farmer 1994, N. V. Rao and A. C. Roy 1970, Roy 1968 and in the references therein). In this note we shall present a procedure to obtain extremal elements of the unit ball of the quasi-normed semilinear space of real-valued semi-Lipschitz functions defined on a quasi-metric space.
MSC 2000. 46A22, 26A16.
Keywords. semi metric spaces, semi Lipschitz real functions.
Keywords. semi metric spaces, semi Lipschitz real functions.
1. INTRODUCTION
Let XX be a nonvoid set. A function d:X rarr[0,oo]d: X \rightarrow[0, \infty] is called a quasi - metric if it satisfies the conditions:
(i) d(x,y)=d(y,x)=0Longleftrightarrow x=yd(x, y)=d(y, x)=0 \Longleftrightarrow x=y
(ii) d(x,y) <= d(x,z)+d(z,y)d(x, y) \leq d(x, z)+d(z, y)
or
(i') d(x,y)=0Longleftrightarrow x=yd(x, y)=0 \Longleftrightarrow x=y
and (ii), for all x,y,z in Xx, y, z \in X. The pair ( X,dX, d ) is called a quasi - metric space.
Remark that dd is not a symmetric function, i.e., it is possible that d(x,y)!=d(y,x)d(x, y) \neq d(y, x) for x,y in Xx, y \in X.
(i) d(x,y)=d(y,x)=0Longleftrightarrow x=yd(x, y)=d(y, x)=0 \Longleftrightarrow x=y
(ii) d(x,y) <= d(x,z)+d(z,y)d(x, y) \leq d(x, z)+d(z, y)
or
(i') d(x,y)=0Longleftrightarrow x=yd(x, y)=0 \Longleftrightarrow x=y
and (ii), for all x,y,z in Xx, y, z \in X. The pair ( X,dX, d ) is called a quasi - metric space.
Remark that dd is not a symmetric function, i.e., it is possible that d(x,y)!=d(y,x)d(x, y) \neq d(y, x) for x,y in Xx, y \in X.
A function f:X rarrRf: X \rightarrow \mathbb{R}, defined on a quasi - metric space ( X,dX, d ) is called semi-Lipschitz if there exists K >= 0K \geq 0 such that
{:(1)f(x)-f(y) <= K*d(x","y)",":}\begin{equation*}
f(x)-f(y) \leq K \cdot d(x, y), \tag{1}
\end{equation*}
for all x,y in Xx, y \in X.
A function f:X rarrRf: X \rightarrow \mathbb{R} is called <= _(d^(-))\leq_{d^{-}}increasing if
a) d(x,y)=0d(x, y)=0 implies f(x)-f(y) <= 0f(x)-f(y) \leq 0
or, equivalently
{:a^('))f(x)-f(y) > 0\left.a^{\prime}\right) f(x)-f(y)>0 implies d(x,y) > 0d(x, y)>0, for all x,y in Xx, y \in X.
Let
A function f:X rarrRf: X \rightarrow \mathbb{R} is called <= _(d^(-))\leq_{d^{-}}increasing if
a) d(x,y)=0d(x, y)=0 implies f(x)-f(y) <= 0f(x)-f(y) \leq 0
or, equivalently
{:a^('))f(x)-f(y) > 0\left.a^{\prime}\right) f(x)-f(y)>0 implies d(x,y) > 0d(x, y)>0, for all x,y in Xx, y \in X.
Let
{:(2)" SLip "X={f:X rarrR∣f" is " <= _(d)"-increasing and "||f||_(X) < oo}",":}\begin{equation*}
\text { SLip } X=\left\{f: X \rightarrow \mathbb{R} \mid f \text { is } \leq_{d} \text {-increasing and }\|f\|_{X}<\infty\right\}, \tag{2}
\end{equation*}
(see [12]), where
{:(3)||f||_(X)=s u p{((f(x)-f(y))vv0)/(d(x,y)):x,y in X,d(x,y)!=0}:}\begin{equation*} \|f\|_{X}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: x, y \in X, d(x, y) \neq 0\right\} \tag{3} \end{equation*}
{:(3)||f||_(X)=s u p{((f(x)-f(y))vv0)/(d(x,y)):x,y in X,d(x,y)!=0}:}\begin{equation*} \|f\|_{X}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: x, y \in X, d(x, y) \neq 0\right\} \tag{3} \end{equation*}
The set SLipXS L i p X defined in (2) is exactly the set of all semi-Lipschitz functions on ( X,dX, d ), and ||f||_(X)\|f\|_{X} defined by (3) is the least semi-Lipschitz constant for ff, i.e.
{:(4)f(x)-f(y) <= ||f||_(X)*d(x","y)","quad x","y in X:}\begin{equation*}
f(x)-f(y) \leq\|f\|_{X} \cdot d(x, y), \quad x, y \in X \tag{4}
\end{equation*}
and every K >= 0K \geq 0, for which the inequality (1) holds, satisfies K >= ||f||_(X)K \geq\|f\|_{X} (see [9] and [12]).
For x_(0)in Xx_{0} \in X be fixed, denote by
{:(5)S" Lip "_(0)X={f in S" Lip "X:f(x_(0))=0}:}\begin{equation*}
S \text { Lip }_{0} X=\left\{f \in S \text { Lip } X: f\left(x_{0}\right)=0\right\} \tag{5}
\end{equation*}
the set of all real-valued semi-Lipschitz functions defined on the quasi-metric space XX which vanish at the fixed point x_(0)in Xx_{0} \in X.
Let VV be a nonvoid set and R^(+)=[0,oo)\mathbb{R}^{+}=[0, \infty). Suppose that on VV is defined an operation
+:V xx V rarr V+: V \times V \rightarrow V
such that ( V,+V,+ ) is an Abelian semigroup, i.e. + satisfies the conditions
(i) (x+y)+z=x+(y+z)(x+y)+z=x+(y+z)
(ii) x+y=y+xx+y=y+x
(iii) 0+x=x(00+x=x(0 is the neutral element of semigroup (V,+))(V,+))
for all x,y,z in Vx, y, z \in V, and an operation
(i) (x+y)+z=x+(y+z)(x+y)+z=x+(y+z)
(ii) x+y=y+xx+y=y+x
(iii) 0+x=x(00+x=x(0 is the neutral element of semigroup (V,+))(V,+))
for all x,y,z in Vx, y, z \in V, and an operation
*:R^(+)xx V rarr V\cdot: \mathbb{R}^{+} \times V \rightarrow V
having the properties
(i) a*(b*x)=(a*b)*x,a,b inR^(+);x in Va \cdot(b \cdot x)=(a \cdot b) \cdot x, a, b \in \mathbb{R}^{+} ; x \in V
(ii) (a+b)*x=(a*x)+(b*x),a,b inR^(+);x in V(a+b) \cdot x=(a \cdot x)+(b \cdot x), a, b \in \mathbb{R}^{+} ; x \in V
(iii) a*(x+y)=a*x+a*y,a inR^(+);x,y in Va \cdot(x+y)=a \cdot x+a \cdot y, a \in \mathbb{R}^{+} ; x, y \in V
(iv) 1*x=x,1inR^(+),x in V1 \cdot x=x, 1 \in \mathbb{R}^{+}, x \in V.
(i) a*(b*x)=(a*b)*x,a,b inR^(+);x in Va \cdot(b \cdot x)=(a \cdot b) \cdot x, a, b \in \mathbb{R}^{+} ; x \in V
(ii) (a+b)*x=(a*x)+(b*x),a,b inR^(+);x in V(a+b) \cdot x=(a \cdot x)+(b \cdot x), a, b \in \mathbb{R}^{+} ; x \in V
(iii) a*(x+y)=a*x+a*y,a inR^(+);x,y in Va \cdot(x+y)=a \cdot x+a \cdot y, a \in \mathbb{R}^{+} ; x, y \in V
(iv) 1*x=x,1inR^(+),x in V1 \cdot x=x, 1 \in \mathbb{R}^{+}, x \in V.
The system ( V,+,*,R^(+)V,+, \cdot, \mathbb{R}^{+}) is called a semi linear space.
The opposite element (if exists) of x in Vx \in V is denoted by -x-x.
A functional ||*||_(V):V rarr[0,oo)\|\cdot\|_{V}: V \rightarrow[0, \infty) defined on a semilinear space ( V,+,*,R^(+)V,+, \cdot, \mathbb{R}^{+}) is called a quasi-norm on VV if it satisfies the conditions:
(i) x,-x in Vx,-x \in V and ||x||_(V)=||-x||_(V)=0Longleftrightarrow x=0\|x\|_{V}=\|-x\|_{V}=0 \Longleftrightarrow x=0
(ii) ||ax||_(V)=a||x||_(V),a inR^(+),x in V\|a x\|_{V}=a\|x\|_{V}, a \in \mathbb{R}^{+}, x \in V
(iii) ||x+y||_(V) <= ||x||_(V)+||y||_(V),x,y in V\|x+y\|_{V} \leq\|x\|_{V}+\|y\|_{V}, x, y \in V.
The opposite element (if exists) of x in Vx \in V is denoted by -x-x.
A functional ||*||_(V):V rarr[0,oo)\|\cdot\|_{V}: V \rightarrow[0, \infty) defined on a semilinear space ( V,+,*,R^(+)V,+, \cdot, \mathbb{R}^{+}) is called a quasi-norm on VV if it satisfies the conditions:
(i) x,-x in Vx,-x \in V and ||x||_(V)=||-x||_(V)=0Longleftrightarrow x=0\|x\|_{V}=\|-x\|_{V}=0 \Longleftrightarrow x=0
(ii) ||ax||_(V)=a||x||_(V),a inR^(+),x in V\|a x\|_{V}=a\|x\|_{V}, a \in \mathbb{R}^{+}, x \in V
(iii) ||x+y||_(V) <= ||x||_(V)+||y||_(V),x,y in V\|x+y\|_{V} \leq\|x\|_{V}+\|y\|_{V}, x, y \in V.
The pair ( V,||*||_(V)V,\|\cdot\|_{V} ) is called a quasi-normed semilinear space (see [5] and [12]).
If XX is a linear space then a functional ||*||_(X):X rarr[0,oo)\|\cdot\|_{X}: X \rightarrow[0, \infty) satisfying the axioms of a quasi-norm is called an asymmetric norm on XX (see [4]).
It is immediate that the functional defined by (3) is a quasi-norm on SLip_(0)XS L i p_{0} X, i.e. the pair ( SLip_(0)X,||*||_(X)S L i p_{0} X,\|\cdot\|_{X} ) is a quasi-normed semilinear space.
If Y sub XY \subset X and x_(0)in Yx_{0} \in Y then one considers the semi-Lipschitz functions on YY which vanish at x_(0)x_{0} and the quasi-normed semilinear space ( SS Lip _(0)Y,||*||_(Y){ }_{0} Y,\|\cdot\|_{Y} ), where ||*||_(Y)\|\cdot\|_{Y} is defined like in (3) with YY instead of XX.
The following extension theorem for semi-Lipschitz functions is similar to Mc Shane's [6] extension theorem for Lipschitz functions.
Theorem 1. [9]. Let ( X,dX, d ) be a quasi-metric space, x_(0)in Xx_{0} \in X fixed and Y sub XY \subset X such that x_(0)in Yx_{0} \in Y. Then every function f in Sf \in S Lip _(0)Y{ }_{0} Y admits at least one extension in SLip_(0)XS L i p_{0} X, i.e. there exists H in SH \in S Lip _(0)X{ }_{0} X such that
{:(6)H|_(Y)=f" and "||H||_(X)=||f||_(Y):}\begin{equation*}
\left.H\right|_{Y}=f \text { and }\|H\|_{X}=\|f\|_{Y} \tag{6}
\end{equation*}
Denote by
{:(7)E_(Y)(f)={H in SLip_(0)X:H|_(Y)=f" and "||H||_(X)=||f||_(Y)}:}\begin{equation*}
E_{Y}(f)=\left\{H \in S \operatorname{Lip}_{0} X:\left.H\right|_{Y}=f \text { and }\|H\|_{X}=\|f\|_{Y}\right\} \tag{7}
\end{equation*}
the nonvoid set of all extensions of f in SLip_(0)Yf \in S L i p_{0} Y which preserve the quasi-norm of ff.
We have shown in [9] that the functions
{:(8)F(x)=i n f_(y in Y){f(y)+||f||_(Y)d(x,y)}","quad x in X:}\begin{equation*}
F(x)=\inf _{y \in Y}\left\{f(y)+\|f\|_{Y} d(x, y)\right\}, \quad x \in X \tag{8}
\end{equation*}
and
{:(9)G(x)=s u p_(y in Y){f(y)-||f||_(Y)d(y,x)}","quad x in X:}\begin{equation*}
G(x)=\sup _{y \in Y}\left\{f(y)-\|f\|_{Y} d(y, x)\right\}, \quad x \in X \tag{9}
\end{equation*}
belong to E_(Y)(f)E_{Y}(f).
Let
Let
{:(10)B_(Y)={f in SLip_(0)Y:||f||_(Y) <= 1}:}\begin{equation*}
B_{Y}=\left\{f \in S \operatorname{Lip}_{0} Y:\|f\|_{Y} \leq 1\right\} \tag{10}
\end{equation*}
be the unit ball of the quasi-normed semilinear space ( SLip_(0)Y,||*||_(Y)S L i p_{0} Y,\|\cdot\|_{Y} ) and let B_(X)B_{X} be the corresponding unit ball of ( SLip_(0)Y,||*||_(X)S L i p_{0} Y,\|\cdot\|_{X} ).
Obviously that f inB_(Y)f \in B_{Y} implies E_(Y)(f)subB_(X)E_{Y}(f) \subset B_{X}.
A subset CC of a semi-linear space ( V,+,*,R^(+)V,+, \cdot, \mathbb{R}^{+}) is called convex if alpha x+(1-alpha)y in C\alpha x+ (1-\alpha) y \in C whenever x,y in Cx, y \in C and alpha in[0,1]\alpha \in[0,1].
A subset CC of a semi-linear space ( V,+,*,R^(+)V,+, \cdot, \mathbb{R}^{+}) is called convex if alpha x+(1-alpha)y in C\alpha x+ (1-\alpha) y \in C whenever x,y in Cx, y \in C and alpha in[0,1]\alpha \in[0,1].
A subset MM of CC is called a face of CC if lambda x+(1-lambda)y in M\lambda x+(1-\lambda) y \in M for x,y in Cx, y \in C and some lambda in(0,1)\lambda \in(0,1) implies x,y in Mx, y \in M. A one-point face of CC is called an extremal element of CC, and the set of all extremal elements of CC is denoted by ext CC.
It is obvious that B_(Y)B_{Y} (respectively B_(X)B_{X} ) is a convex subset of SLip_(0)YS L i p_{0} Y (respectively SLip_(0)XS L i p_{0} X ), and if M subB_(X)M \subset B_{X} is a face, then ||f||_(X)=1\|f\|_{X}=1 for any f in Mf \in M.
Theorem 2. Let ( X,dX, d ) be a quasi-metric space, x_(0)x_{0} a fixed point in XX, and Y sub XY \subset X such that x_(0)in Yx_{0} \in Y. Then:
a) For every f in Sf \in S Lip _(0)Y{ }_{0} Y the set E_(Y)(f)sub SE_{Y}(f) \subset S Lip _(0)X{ }_{0} X is convex;
b) For every H inE_(Y)(f)H \in E_{Y}(f) the inequalities
a) For every f in Sf \in S Lip _(0)Y{ }_{0} Y the set E_(Y)(f)sub SE_{Y}(f) \subset S Lip _(0)X{ }_{0} X is convex;
b) For every H inE_(Y)(f)H \in E_{Y}(f) the inequalities
{:(11)F(x) >= H(x) >= G(x):}\begin{equation*}
F(x) \geq H(x) \geq G(x) \tag{11}
\end{equation*}
hold for all x in Xx \in X, where the functions FF and GG are defined by (8) and (9), respectively;
c) If f inf \in ext B_(Y)B_{Y} then E_(Y)(f)E_{Y}(f) is a face of B_(X)B_{X} and the functions F,GF, G (defined by (8) and (9)) are extremal elements of B_(X)B_{X}.
c) If f inf \in ext B_(Y)B_{Y} then E_(Y)(f)E_{Y}(f) is a face of B_(X)B_{X} and the functions F,GF, G (defined by (8) and (9)) are extremal elements of B_(X)B_{X}.
Proof. a) Let F_(1),F_(2)inE_(Y)(f)F_{1}, F_{2} \in E_{Y}(f) and alpha in(0,1)\alpha \in(0,1). We have
(alphaF_(1)+(1-alpha)F_(2))|_(Y)=alpha f+(1-alpha)f=f\left.\left(\alpha F_{1}+(1-\alpha) F_{2}\right)\right|_{Y}=\alpha f+(1-\alpha) f=f
and
{:[||alphaF_(1)+(1-alpha)F_(2)||_(X) <= alpha||F_(1)||_(X)+(1-alpha)||F_(2)||_(X)],[=alpha||f||_(Y)+(1-alpha)||f||_(Y)=||f||_(Y).]:}\begin{aligned}
\left\|\alpha F_{1}+(1-\alpha) F_{2}\right\|_{X} & \leq \alpha\left\|F_{1}\right\|_{X}+(1-\alpha)\left\|F_{2}\right\|_{X} \\
& =\alpha\|f\|_{Y}+(1-\alpha)\|f\|_{Y}=\|f\|_{Y} .
\end{aligned}
On the other hand
{:[||f||_(Y)=||alpha f+(1-alpha)f||_(Y)],[=|| alphaF_(1)|_(Y)+(1-alpha)F_(2)|_(Y)|| <= ||alphaF_(1)+(1-alpha)F_(2)||_(X)]:}\begin{aligned}
\|f\|_{Y} & =\|\alpha f+(1-\alpha) f\|_{Y} \\
& =\left\|\left.\alpha F_{1}\right|_{Y}+\left.(1-\alpha) F_{2}\right|_{Y}\right\| \leq\left\|\alpha F_{1}+(1-\alpha) F_{2}\right\|_{X}
\end{aligned}
showing that ||alphaF_(1)+(1-alpha)F_(2)||_(X)=||f||_(Y)\left\|\alpha F_{1}+(1-\alpha) F_{2}\right\|_{X}=\|f\|_{Y}. It follows alphaF_(1)+(1-alpha)F_(2)inE_(Y)(f)\alpha F_{1}+(1-\alpha) F_{2} \in E_{Y}(f).
b) Let H inE_(Y)(f)H \in E_{Y}(f) and x in Xx \in X. We have, for any y in Y,H(x)-f(y)=H(x)-H(y) <= ||H||_(X)*d(x,y)=||f||_(Y)*d(x,y)y \in Y, H(x)-f(y)= H(x)-H(y) \leq\|H\|_{X} \cdot d(x, y)=\|f\|_{Y} \cdot d(x, y) so that
b) Let H inE_(Y)(f)H \in E_{Y}(f) and x in Xx \in X. We have, for any y in Y,H(x)-f(y)=H(x)-H(y) <= ||H||_(X)*d(x,y)=||f||_(Y)*d(x,y)y \in Y, H(x)-f(y)= H(x)-H(y) \leq\|H\|_{X} \cdot d(x, y)=\|f\|_{Y} \cdot d(x, y) so that
H(x) <= f(y)+||f||_(Y)d(x,y)H(x) \leq f(y)+\|f\|_{Y} d(x, y)
Taking the infimum with respect to y in Yy \in Y we find
H(x) <= F(x),quad" for all "x in X.H(x) \leq F(x), \quad \text { for all } x \in X .
Also, we have
H(y)-H(x) <= ||H||_(X)*d(y,x)=||f||_(Y)*d(y,x)H(y)-H(x) \leq\|H\|_{X} \cdot d(y, x)=\|f\|_{Y} \cdot d(y, x)
which implies
H(x) >= H(y)-||f||_(Y)*d(y,x)=f(y)-||f||_(Y)*d(y,x)H(x) \geq H(y)-\|f\|_{Y} \cdot d(y, x)=f(y)-\|f\|_{Y} \cdot d(y, x)
Taking the supremum with respect to y in Yy \in Y we get
H(x) >= G(x),quad x in X.H(x) \geq G(x), \quad x \in X .
c) Let f in extB_(Y)f \in \operatorname{ext} B_{Y}. If F_(1),F_(2)inB_(X)F_{1}, F_{2} \in B_{X} and lambda in(0,1)\lambda \in(0,1) are such that lambdaF_(1)+(1-lambda)F_(2)inE_(Y)(f)\lambda F_{1}+ (1-\lambda) F_{2} \in E_{Y}(f) then lambdaF_(1)|_(Y)+(1-lambda)F_(2)|_(Y)=f\left.\lambda F_{1}\right|_{Y}+\left.(1-\lambda) F_{2}\right|_{Y}=f. Since f in extB_(Y)f \in \operatorname{ext} B_{Y} this implies F_(1)|_(Y)=F_(2)|_(Y)=f\left.F_{1}\right|_{Y}=\left.F_{2}\right|_{Y}=f. Obviously that ||F_(1)||_(X)=||F_(2)||_(X)=||f||_(Y)=1\left\|F_{1}\right\|_{X}=\left\|F_{2}\right\|_{X}=\|f\|_{Y}=1, showing that F_(1),F_(2)inE_(Y)(f)F_{1}, F_{2} \in E_{Y}(f). It follows that E_(Y)(f)E_{Y}(f) is a face of B_(X)B_{X}.
We remark that F,GF, G defined by (8), (9) are extremal elements of E_(Y)(f)E_{Y}(f).
Indeed, if H_(1),H_(2)inE_(Y)(f)H_{1}, H_{2} \in E_{Y}(f) and lambda in(0,1)\lambda \in(0,1) are such that lambdaH_(1)+(1-lambda)H_(2)=F\lambda H_{1}+(1-\lambda) H_{2}= F then lambdaH_(1)|_(Y)+(1-lambda)H_(2)|_(Y)=f\left.\lambda H_{1}\right|_{Y}+\left.(1-\lambda) H_{2}\right|_{Y}=f and because f in extB_(Y)f \in \operatorname{ext} B_{Y} it follows H_(1)|_(Y)=H_(2)|_(Y)=f=F|_(Y)\left.H_{1}\right|_{Y}= \left.H_{2}\right|_{Y}=f=\left.F\right|_{Y}.
Indeed, if H_(1),H_(2)inE_(Y)(f)H_{1}, H_{2} \in E_{Y}(f) and lambda in(0,1)\lambda \in(0,1) are such that lambdaH_(1)+(1-lambda)H_(2)=F\lambda H_{1}+(1-\lambda) H_{2}= F then lambdaH_(1)|_(Y)+(1-lambda)H_(2)|_(Y)=f\left.\lambda H_{1}\right|_{Y}+\left.(1-\lambda) H_{2}\right|_{Y}=f and because f in extB_(Y)f \in \operatorname{ext} B_{Y} it follows H_(1)|_(Y)=H_(2)|_(Y)=f=F|_(Y)\left.H_{1}\right|_{Y}= \left.H_{2}\right|_{Y}=f=\left.F\right|_{Y}.
On the other hand lambdaH_(1)(x)+(1-lambda)H_(2)(x)=F(x),quad x in X\lambda H_{1}(x)+(1-\lambda) H_{2}(x)=F(x), \quad x \in X implies
lambda(H_(1)(x)-F(x))+(1-lambda)(H_(2)(x)-F(x))=0,quad x in X\lambda\left(H_{1}(x)-F(x)\right)+(1-\lambda)\left(H_{2}(x)-F(x)\right)=0, \quad x \in X
and because H_(1)(x) <= F(x),H_(2)(x) <= F(x),x in XH_{1}(x) \leq F(x), H_{2}(x) \leq F(x), x \in X and lambda in(0,1)\lambda \in(0,1) it follows
{:[H_(1)(x)=F(x)",",x in X","],[H_(2)(x)=F(x)",",x in X.]:}\begin{array}{ll}
H_{1}(x)=F(x), & x \in X, \\
H_{2}(x)=F(x), & x \in X .
\end{array}
Consequently F in extE_(Y)(f)F \in \operatorname{ext} E_{Y}(f). Analogously one obtains G in extE_(Y)(f)G \in \operatorname{ext} E_{Y}(f).
Now let be given U_(1),U_(2)inB_(X)U_{1}, U_{2} \in B_{X} and lambda in(0,1)\lambda \in(0,1) such that lambdaU_(1)+(1-lambda)U_(2)=F\lambda U_{1}+(1-\lambda) U_{2}=F. Then lambdaU_(1)|_(Y)+(1-lambda)U_(2)|_(Y)=F|_(Y)=f in extB_(Y)\left.\lambda U_{1}\right|_{Y}+\left.(1-\lambda) U_{2}\right|_{Y}=\left.F\right|_{Y}=f \in \operatorname{ext} B_{Y} implies U_(1)|_(Y)=U_(2)|_(Y)=f\left.U_{1}\right|_{Y}=\left.U_{2}\right|_{Y}=f and ||U_(1)|_(Y)||_(Y)=||U_(2)|_(Y)||_(Y)=||f||=1\left\|\left.U_{1}\right|_{Y}\right\|_{Y}=\left\|\left.U_{2}\right|_{Y}\right\|_{Y}=\|f\|=1 implies ||U_(1)||_(X)=||U_(2)||_(X)=1\left\|U_{1}\right\|_{X}=\left\|U_{2}\right\|_{X}=1.
Now let be given U_(1),U_(2)inB_(X)U_{1}, U_{2} \in B_{X} and lambda in(0,1)\lambda \in(0,1) such that lambdaU_(1)+(1-lambda)U_(2)=F\lambda U_{1}+(1-\lambda) U_{2}=F. Then lambdaU_(1)|_(Y)+(1-lambda)U_(2)|_(Y)=F|_(Y)=f in extB_(Y)\left.\lambda U_{1}\right|_{Y}+\left.(1-\lambda) U_{2}\right|_{Y}=\left.F\right|_{Y}=f \in \operatorname{ext} B_{Y} implies U_(1)|_(Y)=U_(2)|_(Y)=f\left.U_{1}\right|_{Y}=\left.U_{2}\right|_{Y}=f and ||U_(1)|_(Y)||_(Y)=||U_(2)|_(Y)||_(Y)=||f||=1\left\|\left.U_{1}\right|_{Y}\right\|_{Y}=\left\|\left.U_{2}\right|_{Y}\right\|_{Y}=\|f\|=1 implies ||U_(1)||_(X)=||U_(2)||_(X)=1\left\|U_{1}\right\|_{X}=\left\|U_{2}\right\|_{X}=1.
It follows that U_(1),U_(2)inE_(Y)(f)U_{1}, U_{2} \in E_{Y}(f) and because F in extE_(Y)(f)F \in \operatorname{ext} E_{Y}(f) one obtains U_(1)=U_(2)=FU_{1}=U_{2}=F. It follows that F in extB_(X)F \in \operatorname{ext} B_{X} and, analogously G in extB_(X)G \in \operatorname{ext} B_{X}.
Remarks. 1^(@)1^{\circ}. The reverse implication in c) is also true: if E_(Y)(f)E_{Y}(f) is a face of B_(X)B_{X} then f inB_(Y)f \in B_{Y}.
Indeed, if ||f||_(Y)=1\|f\|_{Y}=1 but f!in extB_(Y)f \notin \operatorname{ext} B_{Y}, then there exist f_(1),f_(2)inB_(Y),f_(1)!=f_(2)f_{1}, f_{2} \in B_{Y}, f_{1} \neq f_{2}, and lambda in(0,1)\lambda \in(0,1) such that lambdaf_(1)+(1-lambda)f_(2)=f\lambda f_{1}+(1-\lambda) f_{2}=f.
Let F_(1)^(')inE_(Y)(f_(1))F_{1}^{\prime} \in E_{Y}\left(f_{1}\right) and F_(2)^(')inE_(Y)(f_(2))F_{2}^{\prime} \in E_{Y}\left(f_{2}\right). Because
lambdaF_(1)^(')|_(Y)+(1-lambda)F_(2)^(')|_(Y)=f\left.\lambda F_{1}^{\prime}\right|_{Y}+\left.(1-\lambda) F_{2}^{\prime}\right|_{Y}=f
and
1=|| lambdaF_(1)^(')|_(Y)+(1-lambda)F_(2)^(')|_(Y)|| <= ||lambdaF_(1)^(')+(1-lambda)F_(2)^(')||_(X) <= 11=\left\|\left.\lambda F_{1}^{\prime}\right|_{Y}+\left.(1-\lambda) F_{2}^{\prime}\right|_{Y}\right\| \leq\left\|\lambda F_{1}^{\prime}+(1-\lambda) F_{2}^{\prime}\right\|_{X} \leq 1
we have
||lambdaF_(1)^(')+(1-lambda)F_(2)^(')||_(X)=1,\left\|\lambda F_{1}^{\prime}+(1-\lambda) F_{2}^{\prime}\right\|_{X}=1,
showing that lambdaF_(1)^(')+(1-lambda)F_(2)^(')inE_(Y)(f)\lambda F_{1}^{\prime}+(1-\lambda) F_{2}^{\prime} \in E_{Y}(f). Since F_(1)^(')|_(Y)=f_(1)!=f_(2)=F_(2)^(')|_(Y)\left.F_{1}^{\prime}\right|_{Y}=f_{1} \neq f_{2}=\left.F_{2}^{\prime}\right|_{Y} it follows that E_(Y)(f)E_{Y}(f) is not a face of B_(X)B_{X}.
2^(@)2^{\circ}. The assertion c) from Theorem 2 gives us a way to obtain extremal elements of B_(X)B_{X}, namely as the extensions (8) and (9) of extremal elements of B_(Y)B_{Y}.
2^(@)2^{\circ}. The assertion c) from Theorem 2 gives us a way to obtain extremal elements of B_(X)B_{X}, namely as the extensions (8) and (9) of extremal elements of B_(Y)B_{Y}.
Example. Consider the quasi metric space (R,d)(\mathbb{R}, d), where R\mathbb{R} is the set of real numbers and
d(x,y)={[x-y," if ",x >= y],[1," if ",x < y]:}d(x, y)=\left\{\begin{array}{cll}
x-y & \text { if } & x \geq y \\
1 & \text { if } & x<y
\end{array}\right.
For Y={0,1,2}Y=\{0,1,2\} and x_(0)=0x_{0}=0 consider the semilinear spaces SLip_(0)YS L i p_{0} Y and SLip_(0)XS L i p_{0} X equipped with the quasi-norms of the type (3).
The function f(y)=y,y in{0,1,2}=Yf(y)=y, y \in\{0,1,2\}=Y is an extremal element of B_(Y)B_{Y}. Observe that for any h inB_(Y)h \in B_{Y} we have h(1) <= f(1)=1h(1) \leq f(1)=1 and h(2) <= f(2)=2h(2) \leq f(2)=2, because, if contrary, i.e. h(1) > f(1)h(1)>f(1) or h(2) > f(2)h(2)>f(2), then ||h||_(Y) > 1\|h\|_{Y}>1. If f_(1),f_(2)inB_(Y)f_{1}, f_{2} \in B_{Y} and alpha in(0,1)\alpha \in(0,1) are such that alphaf_(1)+(1-alpha)f_(2)=f\alpha f_{1}+(1-\alpha) f_{2}=f then, taking into account the relations f_(i)(0)=f(0)=0,f_(i)(1) <= f(1),f_(i)(2) <= f(2),i=1,2f_{i}(0)=f(0)=0, f_{i}(1) \leq f(1), f_{i}(2) \leq f(2), i=1,2, we get alpha(f_(1)-f)+(1-alpha)(f_(2)-f)=0\alpha\left(f_{1}-f\right)+(1-\alpha)\left(f_{2}-f\right)=0 implying f_(1)=f_(2)=ff_{1}=f_{2}=f.
In this case the extensions FF and GG, given by (8) and (9), are
F(x)={[1","," for "x in(-oo","0)],[x","," for "x in[0","+oo)]" respectively "G(x)={[x","," for "x in(-oo","2]],[1","," for "x in(2","+oo)]:}F(x)=\left\{\begin{array}{ll}
1, & \text { for } x \in(-\infty, 0) \\
x, & \text { for } x \in[0,+\infty)
\end{array} \text { respectively } G(x)= \begin{cases}x, & \text { for } x \in(-\infty, 2] \\
1, & \text { for } x \in(2,+\infty)\end{cases}\right.
and they are extremal elements of B_(X)B_{X}.
REFERENCES
[1] Cobzaş, S. and Mustăţa, C., Norm preserving extension of convex Lipschitz functions, J. Approx. Theory, 24, pp. 555-564, 1978.
[2] Cobzaş, S., Extreme points in Banach spaces of Lipschitz functions, Mathematica, 31 (54), pp. 25-33, 1989.
[3] Farmer, J. D., Extreme points of the unit ball of the space of Lipschitz functions, Proc. Amer. Math. Soc., 121, no. 3, pp. 807-813, 1994.
[4] Krein, M. G. and A. A. Nudel'man, The Markov Moment Problem and Extremum Problems, Nauka, Moscow, 1973, (in Russian).
[5] Koppermann, R. D., All topologies come from generalized metrics, Amer. Math. Monthly, 95, pp. 89-97, 1988.
[6] McShane, E. J., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837842, 1934.
[7] MustĂta, C., Best approximation and unique extension of Lipschitz functions, J. Approx. Theory, 19, pp. 222-230, 1977.
[8] Mustăţa, C., Uniquenness of the extension of semi-Lipschitz functions on quasi-metric spaces, Bull. Şt. Univ. Baia Mare, Mat.-Inf., XVI, no. 2, pp. 207-212, 2000.
[9] MustĂţA, C., Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numér. Théor. Approx., 2001 (to appear). 주
[10] Phelps, R. R., Uniqueness of Hahn-Banach extension and unique best approximation, Trans. Amer. Math. Soc., 95, pp. 238-255, 1960.
[11] Rao, N. V. and Roy, A. K., Extreme Lipschitz functions, Math. Ann., 189, pp. 26-46, 1970.
[12] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric space, J. Approx. Theory, 103, pp. 293-301, 2000.
[13] Roy, A. K., Extreme points and linear isometries of Banach spaces of Lipschitz functions, Canad. J. Math., 20, pp. 1150-1164, 1968.
[14] Wells, J. H. and Williams, L. R., Embedings and Extensions in Analysis, SpringerVerlag, Berlin, 1975.
[2] Cobzaş, S., Extreme points in Banach spaces of Lipschitz functions, Mathematica, 31 (54), pp. 25-33, 1989.
[3] Farmer, J. D., Extreme points of the unit ball of the space of Lipschitz functions, Proc. Amer. Math. Soc., 121, no. 3, pp. 807-813, 1994.
[4] Krein, M. G. and A. A. Nudel'man, The Markov Moment Problem and Extremum Problems, Nauka, Moscow, 1973, (in Russian).
[5] Koppermann, R. D., All topologies come from generalized metrics, Amer. Math. Monthly, 95, pp. 89-97, 1988.
[6] McShane, E. J., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837842, 1934.
[7] MustĂta, C., Best approximation and unique extension of Lipschitz functions, J. Approx. Theory, 19, pp. 222-230, 1977.
[8] Mustăţa, C., Uniquenness of the extension of semi-Lipschitz functions on quasi-metric spaces, Bull. Şt. Univ. Baia Mare, Mat.-Inf., XVI, no. 2, pp. 207-212, 2000.
[9] MustĂţA, C., Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numér. Théor. Approx., 2001 (to appear). 주
[10] Phelps, R. R., Uniqueness of Hahn-Banach extension and unique best approximation, Trans. Amer. Math. Soc., 95, pp. 238-255, 1960.
[11] Rao, N. V. and Roy, A. K., Extreme Lipschitz functions, Math. Ann., 189, pp. 26-46, 1970.
[12] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric space, J. Approx. Theory, 103, pp. 293-301, 2000.
[13] Roy, A. K., Extreme points and linear isometries of Banach spaces of Lipschitz functions, Canad. J. Math., 20, pp. 1150-1164, 1968.
[14] Wells, J. H. and Williams, L. R., Embedings and Extensions in Analysis, SpringerVerlag, Berlin, 1975.
Received by the editors: September 26, 2001.
- *"T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, 3400 Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro.