Return to Article Details On the extremal semi-Lipschitz functions

REVUE D'ANALYSE NUMÉRIQUE ET DE THÉORIE DE L'APPROXIMATION

Rev. Anal. Numér. Théor. Approx., vol. 31 (2002) no. 1, pp. 103-108 ictp.acad.ro/jnaat

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.

1. INTRODUCTION

Let X X XXX be a nonvoid set. A function d : X → [ 0 , ∞ ] d : X → [ 0 , ∞ ] d:X rarr[0,oo]d: X \rightarrow[0, \infty]d:X→[0,∞] is called a
quasi - metric if it satisfies the conditions:
(i) d ( x , y ) = d ( y , x ) = 0 ⟺ x = y d ( x , y ) = d ( y , x ) = 0 ⟺ x = y d(x,y)=d(y,x)=0Longleftrightarrow x=yd(x, y)=d(y, x)=0 \Longleftrightarrow x=yd(x,y)=d(y,x)=0⟺x=y
(ii) d ( x , y ) ≤ d ( x , z ) + d ( z , y ) d ( x , y ) ≤ d ( x , z ) + d ( z , y ) d(x,y) <= d(x,z)+d(z,y)d(x, y) \leq d(x, z)+d(z, y)d(x,y)≤d(x,z)+d(z,y)
or
(i') d ( x , y ) = 0 ⟺ x = y d ( x , y ) = 0 ⟺ x = y d(x,y)=0Longleftrightarrow x=yd(x, y)=0 \Longleftrightarrow x=yd(x,y)=0⟺x=y
and (ii), for all x , y , z ∈ X x , y , z ∈ X x,y,z in Xx, y, z \in Xx,y,z∈X. The pair ( X , d X , d X,dX, dX,d ) is called a quasi - metric space.
Remark that d d ddd is not a symmetric function, i.e., it is possible that d ( x , y ) ≠ d ( y , x ) d ( x , y ) ≠ d ( y , x ) d(x,y)!=d(y,x)d(x, y) \neq d(y, x)d(x,y)≠d(y,x) for x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X.
A function f : X → R f : X → R f:X rarrRf: X \rightarrow \mathbb{R}f:X→R, defined on a quasi - metric space ( X , d X , d X,dX, dX,d ) is called semi-Lipschitz if there exists K ≥ 0 K ≥ 0 K >= 0K \geq 0K≥0 such that
(1) f ( x ) − f ( y ) ≤ K ⋅ d ( x , y ) , (1) f ( x ) − f ( y ) ≤ K ⋅ d ( x , y ) , {:(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*}(1)f(x)−f(y)≤K⋅d(x,y),
for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X.
A function f : X → R f : X → R f:X rarrRf: X \rightarrow \mathbb{R}f:X→R is called ≤ d − ≤ d − <= _(d^(-))\leq_{d^{-}}≤d−increasing if
a) d ( x , y ) = 0 d ( x , y ) = 0 d(x,y)=0d(x, y)=0d(x,y)=0 implies f ( x ) − f ( y ) ≤ 0 f ( x ) − f ( y ) ≤ 0 f(x)-f(y) <= 0f(x)-f(y) \leq 0f(x)−f(y)≤0
or, equivalently
a ′ ) f ( x ) − f ( y ) > 0 a ′ f ( x ) − f ( y ) > 0 {:a^('))f(x)-f(y) > 0\left.a^{\prime}\right) f(x)-f(y)>0a′)f(x)−f(y)>0 implies d ( x , y ) > 0 d ( x , y ) > 0 d(x,y) > 0d(x, y)>0d(x,y)>0, for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X.
Let
(2) SLip X = { f : X → R ∣ f is ≤ d -increasing and ‖ f ‖ X < ∞ } (2) SLip X = f : X → R ∣ f  is  ≤ d -increasing and  ‖ f ‖ X < ∞ {:(2)SLip X={f:X rarrR∣f" is " <= _(d)"-increasing and "||f||_(X) < oo}:}\begin{equation*} \operatorname{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*}(2)SLipX={f:X→R∣f is ≤d-increasing and ‖f‖X<∞}
(see [12]), where
(3) ‖ f ‖ X = sup { ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) : x , y ∈ X , d ( x , y ) ≠ 0 } . (3) ‖ f ‖ X = sup ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) : x , y ∈ X , d ( x , y ) ≠ 0 . {:(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=sup{(f(x)−f(y))∨0d(x,y):x,y∈X,d(x,y)≠0}.
The set S Lip X S Lip X S Lip XS \operatorname{Lip} XSLipX defined in (2) is exactly the set of all semi-Lipschitz functions on ( X , d X , d X,dX, dX,d ), and ‖ f ‖ X ‖ f ‖ X ||f||_(X)\|f\|_{X}‖f‖X defined by (3) is the least semi-Lipschitz constant for f f fff, i.e.
(4) f ( x ) − f ( y ) ≤ ‖ f ‖ X ⋅ d ( x , y ) , x , y ∈ X (4) f ( x ) − f ( y ) ≤ ‖ f ‖ X ⋅ d ( x , y ) , x , y ∈ X {:(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*}(4)f(x)−f(y)≤‖f‖X⋅d(x,y),x,y∈X
and every K ≥ 0 K ≥ 0 K >= 0K \geq 0K≥0, for which the inequality (1) holds, satisfies K ≥ ‖ f ‖ X K ≥ ‖ f ‖ X K >= ||f||_(X)K \geq\|f\|_{X}K≥‖f‖X (see [9] and [12]).
For x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X be fixed, denote by
(5) S L i p 0 X = { f ∈ S L i p X : f ( x 0 ) = 0 } (5) S L i p 0 X = f ∈ S L i p X : f x 0 = 0 {:(5)SLip_(0)X={f in SLipX:f(x_(0))=0}:}\begin{equation*} S L i p_{0} X=\left\{f \in S L i p X: f\left(x_{0}\right)=0\right\} \tag{5} \end{equation*}(5)SLip0X={f∈SLipX:f(x0)=0}
the set of all real-valued semi-Lipschitz functions defined on the quasi-metric space X X XXX which vanish at the fixed point x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X.
Let V V VVV be a nonvoid set and R + = [ 0 , ∞ ) R + = [ 0 , ∞ ) R^(+)=[0,oo)\mathbb{R}^{+}=[0, \infty)R+=[0,∞). Suppose that on V V VVV is defined an operation
+ : V × V → V + : V × V → V +:V xx V rarr V+: V \times V \rightarrow V+:V×V→V
such that ( V , + ) ( V , + ) (V,+)(V,+)(V,+) is an Abelian semigroup, i.e. + satisfies the conditions
(i) ( x + y ) + z = x + ( y + z ) ( x + y ) + z = x + ( y + z ) (x+y)+z=x+(y+z)(x+y)+z=x+(y+z)(x+y)+z=x+(y+z)
(ii) x + y = y + x x + y = y + x x+y=y+xx+y=y+xx+y=y+x
(iii) 0 + x = x 0 + x = x 0+x=x0+x=x0+x=x (0 is the neutral element of semigroup ( V , + ) ( V , + ) (V,+)(V,+)(V,+) )
for all x , y , z ∈ V x , y , z ∈ V x,y,z in Vx, y, z \in Vx,y,z∈V, and an operation
⋅ : R + × V → V ⋅ : R + × V → V *:R^(+)xx V rarr V\cdot: \mathbb{R}^{+} \times V \rightarrow V⋅:R+×V→V
having the properties
(i) a ⋅ ( b ⋅ x ) = ( a ⋅ b ) ⋅ x , a , b ∈ R + ; x ∈ V a ⋅ ( b ⋅ x ) = ( a ⋅ b ) ⋅ x , a , b ∈ R + ; x ∈ V 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 Va⋅(b⋅x)=(a⋅b)⋅x,a,b∈R+;x∈V
(ii) ( a + b ) ⋅ x = ( a ⋅ x ) + ( b ⋅ x ) , a , b ∈ R + ; x ∈ V ( a + b ) ⋅ x = ( a ⋅ x ) + ( b ⋅ x ) , a , b ∈ R + ; x ∈ V (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(a+b)⋅x=(a⋅x)+(b⋅x),a,b∈R+;x∈V
(iii) a ⋅ ( x + y ) = a ⋅ x + a ⋅ y , a ∈ R + ; x , y ∈ V a ⋅ ( x + y ) = a ⋅ x + a ⋅ y , a ∈ R + ; x , y ∈ V 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 Va⋅(x+y)=a⋅x+a⋅y,a∈R+;x,y∈V
(iv) 1 ⋅ x = x , 1 ∈ R + , x ∈ V 1 ⋅ x = x , 1 ∈ R + , x ∈ V 1*x=x,1inR^(+),x in V1 \cdot x=x, 1 \in \mathbb{R}^{+}, x \in V1⋅x=x,1∈R+,x∈V.
The system ( V , + , ⋅ , R + V , + , ⋅ , R + V,+,*,R^(+)V,+, \cdot, \mathbb{R}^{+}V,+,⋅,R+) is called a semi linear space.
The opposite element (if exists) of x ∈ V x ∈ V x in Vx \in Vx∈V is denoted by − x − x -x-x−x.
A functional ‖ ⋅ ‖ V : V → [ 0 , ∞ ) ‖ ⋅ ‖ V : V → [ 0 , ∞ ) ||*||_(V):V rarr[0,oo)\|\cdot\|_{V}: V \rightarrow[0, \infty)‖⋅‖V:V→[0,∞) defined on a semilinear space ( V , + , ⋅ , R + V , + , ⋅ , R + V,+,*,R^(+)V,+, \cdot, \mathbb{R}^{+}V,+,⋅,R+) is called a quasi-norm on V V VVV if it satisfies the conditions:
(i) x , − x ∈ V x , − x ∈ V x,-x in Vx,-x \in Vx,−x∈V and ‖ x ‖ V = ‖ − x ‖ V = 0 ⟺ x = 0 ‖ x ‖ V = ‖ − x ‖ V = 0 ⟺ x = 0 ||x||_(V)=||-x||_(V)=0Longleftrightarrow x=0\|x\|_{V}=\|-x\|_{V}=0 \Longleftrightarrow x=0‖x‖V=‖−x‖V=0⟺x=0
(ii) ‖ a x ‖ V = a ‖ x ‖ V , a ∈ R + , x ∈ V ‖ a x ‖ V = a ‖ x ‖ V , a ∈ R + , x ∈ V ||ax||_(V)=a||x||_(V),a inR^(+),x in V\|a x\|_{V}=a\|x\|_{V}, a \in \mathbb{R}^{+}, x \in V‖ax‖V=a‖x‖V,a∈R+,x∈V
(iii) ‖ x + y ‖ V ≤ ‖ x ‖ V + ‖ y ‖ V , x , y ∈ V ‖ x + y ‖ V ≤ ‖ x ‖ V + ‖ y ‖ V , x , y ∈ V ||x+y||_(V) <= ||x||_(V)+||y||_(V),x,y in V\|x+y\|_{V} \leq\|x\|_{V}+\|y\|_{V}, x, y \in V‖x+y‖V≤‖x‖V+‖y‖V,x,y∈V.
The pair ( V , ‖ ⋅ ‖ V V , ‖ ⋅ ‖ V V,||*||_(V)V,\|\cdot\|_{V}V,‖⋅‖V ) is called a quasi-normed semilinear space (see [5] and [12]).
If X X XXX is a linear space then a functional ‖ ⋅ ‖ X : X → [ 0 , ∞ ) ‖ ⋅ ‖ X : X → [ 0 , ∞ ) ||*||_(X):X rarr[0,oo)\|\cdot\|_{X}: X \rightarrow[0, \infty)‖⋅‖X:X→[0,∞) satisfying the axioms of a quasi-norm is called an asymmetric norm on X X XXX (see [4]).
It is immediate that the functional defined by (3) is a quasi-norm on SLip 0 X SLip 0 X SLip_(0)X\operatorname{SLip}_{0} XSLip0X, i.e. the pair ( S L i p 0 X , ‖ ⋅ ‖ X S L i p 0 X , ‖ ⋅ ‖ X SLip_(0)X,||*||_(X)S L i p_{0} X,\|\cdot\|_{X}SLip0X,‖⋅‖X ) is a quasi-normed semilinear space.
If Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X and x 0 ∈ Y x 0 ∈ Y x_(0)in Yx_{0} \in Yx0∈Y then one considers the semi-Lipschitz functions on Y Y YYY which vanish at x 0 x 0 x_(0)x_{0}x0 and the quasi-normed semilinear space (SLip Y , ‖ ⋅ ‖ Y Y , ‖ ⋅ ‖ Y Y,||*||_(Y)Y,\|\cdot\|_{Y}Y,‖⋅‖Y ), where ‖ ⋅ ‖ Y ‖ ⋅ ‖ Y ||*||_(Y)\|\cdot\|_{Y}‖⋅‖Y is defined like in (3) with Y Y YYY instead of X X XXX.
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 , d ) ( X , d ) (X,d)(X, d)(X,d) be a quasi-metric space, x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X fixed and Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X such that x 0 ∈ Y x 0 ∈ Y x_(0)in Yx_{0} \in Yx0∈Y. Then every function f ∈ SLip 0 Y f ∈ SLip 0 Y f inSLip_(0)Yf \in \operatorname{SLip}_{0} Yf∈SLip0Y admits at least one extension in SLip 0 X SLip 0 X SLip_(0)X\operatorname{SLip}_{0} XSLip0X, i.e. there exists H ∈ SLip 0 X H ∈ SLip 0 X H inSLip_(0)XH \in \operatorname{SLip}_{0} XH∈SLip0X such that
(6) H | Y = f and ‖ H ‖ X = ‖ f ‖ Y (6) H Y = f  and  ‖ H ‖ X = ‖ f ‖ Y {:(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*}(6)H|Y=f and ‖H‖X=‖f‖Y
Denote by
(7) E Y ( f ) = { H ∈ S L i p 0 X : H | Y = f and ‖ H ‖ X = ‖ f ‖ Y } (7) E Y ( f ) = H ∈ S L i p 0 X : H Y = f  and  ‖ H ‖ X = ‖ f ‖ Y {:(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 L i p_{0} X:\left.H\right|_{Y}=f \text { and }\|H\|_{X}=\|f\|_{Y}\right\} \tag{7} \end{equation*}(7)EY(f)={H∈SLip0X:H|Y=f and ‖H‖X=‖f‖Y}
the nonvoid set of all extensions of f ∈ SLip 0 Y f ∈ SLip 0 Y f inSLip_(0)Yf \in \operatorname{SLip}_{0} Yf∈SLip0Y which preserve the quasi-norm of f f fff.
We have shown in [9] that the functions
(8) F ( x ) = inf y ∈ Y { f ( y ) + ‖ f ‖ Y d ( x , y ) } , x ∈ X (8) F ( x ) = inf y ∈ Y   f ( y ) + ‖ f ‖ Y d ( x , y ) , x ∈ X {:(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*}(8)F(x)=infy∈Y{f(y)+‖f‖Yd(x,y)},x∈X
and
(9) G ( x ) = sup y ∈ Y { f ( y ) − ‖ f ‖ Y d ( y , x ) } , x ∈ X (9) G ( x ) = sup y ∈ Y   f ( y ) − ‖ f ‖ Y d ( y , x ) , x ∈ X {:(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*}(9)G(x)=supy∈Y{f(y)−‖f‖Yd(y,x)},x∈X
belong to E Y ( f ) E Y ( f ) E_(Y)(f)E_{Y}(f)EY(f).
Let
(10) B Y = { f ∈ S L i p 0 Y : ‖ f ‖ Y ≤ 1 } (10) B Y = f ∈ S L i p 0 Y : ‖ f ‖ Y ≤ 1 {:(10)B_(Y)={f in SLip_(0)Y:||f||_(Y) <= 1}:}\begin{equation*} B_{Y}=\left\{f \in S L i p_{0} Y:\|f\|_{Y} \leq 1\right\} \tag{10} \end{equation*}(10)BY={f∈SLip0Y:‖f‖Y≤1}
be the unit ball of the quasi-normed semilinear space ( SLip 0 Y , ‖ ⋅ ‖ Y SLip 0 Y , ‖ ⋅ ‖ Y SLip_(0)Y,||*||_(Y)\operatorname{SLip}_{0} Y,\|\cdot\|_{Y}SLip0Y,‖⋅‖Y ) and let B X B X B_(X)B_{X}BX be the corresponding unit ball of ( SLip 0 Y , ‖ ⋅ ‖ X SLip 0 Y , ‖ ⋅ ‖ X SLip_(0)Y,||*||_(X)\operatorname{SLip}_{0} Y,\|\cdot\|_{X}SLip0Y,‖⋅‖X ).
Obviously that f ∈ B Y f ∈ B Y f inB_(Y)f \in B_{Y}f∈BY implies E Y ( f ) ⊂ B X E Y ( f ) ⊂ B X E_(Y)(f)subB_(X)E_{Y}(f) \subset B_{X}EY(f)⊂BX.
A subset C C CCC of a semi-linear space ( V , + , ⋅ , R + V , + , ⋅ , R + V,+,*,R^(+)V,+, \cdot, \mathbb{R}^{+}V,+,⋅,R+) is called convex if α x + ( 1 − α ) y ∈ C α x + ( 1 − α ) y ∈ C alpha x+(1-alpha)y in C\alpha x+ (1-\alpha) y \in Cαx+(1−α)y∈C whenever x , y ∈ C x , y ∈ C x,y in Cx, y \in Cx,y∈C and α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1].
A subset M M MMM of C C CCC is called a face of C C CCC if λ x + ( 1 − λ ) y ∈ M λ x + ( 1 − λ ) y ∈ M lambda x+(1-lambda)y in M\lambda x+(1-\lambda) y \in Mλx+(1−λ)y∈M for x , y ∈ C x , y ∈ C x,y in Cx, y \in Cx,y∈C and some λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)λ∈(0,1) implies x , y ∈ M x , y ∈ M x,y in Mx, y \in Mx,y∈M. A one-point face of C C CCC is called an extremal element of C C CCC, and the set of all extremal elements of C C CCC is denoted by ext C C CCC.
It is obvious that B Y B Y B_(Y)B_{Y}BY (respectively B X B X B_(X)B_{X}BX ) is a convex subset of SLip 0 Y SLip 0 Y SLip_(0)Y\operatorname{SLip}_{0} YSLip0Y (respectively S L i p 0 X S L i p 0 X SLip_(0)XS L i p_{0} XSLip0X ), and if M ⊂ B X M ⊂ B X M subB_(X)M \subset B_{X}M⊂BX is a face, then ‖ f ‖ X = 1 ‖ f ‖ X = 1 ||f||_(X)=1\|f\|_{X}=1‖f‖X=1 for any f ∈ M f ∈ M f in Mf \in Mf∈M.
Theorem 2. Let ( X , d ) ( X , d ) (X,d)(X, d)(X,d) be a quasi-metric space, x 0 x 0 x_(0)x_{0}x0 a fixed point in X X XXX, and Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X such that x 0 ∈ Y x 0 ∈ Y x_(0)in Yx_{0} \in Yx0∈Y. Then:
a) For every f ∈ S L i p 0 Y f ∈ S L i p 0 Y f in SLip_(0)Yf \in S L i p_{0} Yf∈SLip0Y the set E Y ( f ) ⊂ S L i p 0 X E Y ( f ) ⊂ S L i p 0 X E_(Y)(f)sub SLip_(0)XE_{Y}(f) \subset S L i p_{0} XEY(f)⊂SLip0X is convex;
b) For every H ∈ E Y ( f ) H ∈ E Y ( f ) H inE_(Y)(f)H \in E_{Y}(f)H∈EY(f) the inequalities
(11) F ( x ) ≥ H ( x ) ≥ G ( x ) (11) F ( x ) ≥ H ( x ) ≥ G ( x ) {:(11)F(x) >= H(x) >= G(x):}\begin{equation*} F(x) \geq H(x) \geq G(x) \tag{11} \end{equation*}(11)F(x)≥H(x)≥G(x)
hold for all x ∈ X x ∈ X x in Xx \in Xx∈X, where the functions F F FFF and G G GGG are defined by (8) and (9), respectively;
c) If f ∈ ext B Y f ∈ ext B Y f in extB_(Y)f \in \operatorname{ext} B_{Y}f∈extBY then E Y ( f ) E Y ( f ) E_(Y)(f)E_{Y}(f)EY(f) is a face of B X B X B_(X)B_{X}BX and the functions F , G F , G F,GF, GF,G (defined by (8) and (9)) are extremal elements of B X B X B_(X)B_{X}BX.
Proof. a) Let F 1 , F 2 ∈ E Y ( f ) F 1 , F 2 ∈ E Y ( f ) F_(1),F_(2)inE_(Y)(f)F_{1}, F_{2} \in E_{Y}(f)F1,F2∈EY(f) and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1). We have
( α F 1 + ( 1 − α ) F 2 ) | Y = α f + ( 1 − α ) f = f α F 1 + ( 1 − α ) F 2 Y = α f + ( 1 − α ) f = f (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(αF1+(1−α)F2)|Y=αf+(1−α)f=f
and
‖ α F 1 + ( 1 − α ) F 2 ‖ X ≤ α ‖ F 1 ‖ X + ( 1 − α ) ‖ F 2 ‖ X = α ‖ f ‖ Y + ( 1 − α ) ‖ f ‖ Y = ‖ f ‖ Y α F 1 + ( 1 − α ) F 2 X ≤ α F 1 X + ( 1 − α ) F 2 X = α ‖ f ‖ Y + ( 1 − α ) ‖ f ‖ Y = ‖ f ‖ Y {:[||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}‖αF1+(1−α)F2‖X≤α‖F1‖X+(1−α)‖F2‖X=α‖f‖Y+(1−α)‖f‖Y=‖f‖Y
On the other hand
‖ f ‖ Y = ‖ α f + ( 1 − α ) f ‖ Y = ‖ α F 1 | Y + ( 1 − α ) F 2 | Y ‖ ≤ ‖ α F 1 + ( 1 − α ) F 2 ‖ X ‖ f ‖ Y = ‖ α f + ( 1 − α ) f ‖ Y = α F 1 Y + ( 1 − α ) F 2 Y ≤ α F 1 + ( 1 − α ) F 2 X {:[||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}‖f‖Y=‖αf+(1−α)f‖Y=‖αF1|Y+(1−α)F2|Y‖≤‖αF1+(1−α)F2‖X
showing that ‖ α F 1 + ( 1 − α ) F 2 ‖ X = ‖ f ‖ Y α F 1 + ( 1 − α ) F 2 X = ‖ f ‖ Y ||alphaF_(1)+(1-alpha)F_(2)||_(X)=||f||_(Y)\left\|\alpha F_{1}+(1-\alpha) F_{2}\right\|_{X}=\|f\|_{Y}‖αF1+(1−α)F2‖X=‖f‖Y. It follows α F 1 + ( 1 − α ) F 2 ∈ E Y ( f ) α F 1 + ( 1 − α ) F 2 ∈ E Y ( f ) alphaF_(1)+(1-alpha)F_(2)inE_(Y)(f)\alpha F_{1}+(1-\alpha) F_{2} \in E_{Y}(f)αF1+(1−α)F2∈EY(f).
b) Let H ∈ E Y ( f ) H ∈ E Y ( f ) H inE_(Y)(f)H \in E_{Y}(f)H∈EY(f) and x ∈ X x ∈ X x in Xx \in Xx∈X. We have, for any y ∈ Y , H ( x ) − f ( y ) = H ( x ) − H ( y ) ≤ ‖ H ‖ X ⋅ d ( x , y ) = ‖ f ‖ Y ⋅ d ( x , y ) y ∈ 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) <= ||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)y∈Y,H(x)−f(y)=H(x)−H(y)≤‖H‖X⋅d(x,y)=‖f‖Y⋅d(x,y) so that
H ( x ) ≤ f ( y ) + ‖ f ‖ Y d ( x , y ) H ( x ) ≤ f ( y ) + ‖ f ‖ Y d ( x , y ) H(x) <= f(y)+||f||_(Y)d(x,y)H(x) \leq f(y)+\|f\|_{Y} d(x, y)H(x)≤f(y)+‖f‖Yd(x,y)
Taking the infimum with respect to y ∈ Y y ∈ Y y in Yy \in Yy∈Y we find
H ( x ) ≤ F ( x ) , for all x ∈ X . H ( x ) ≤ F ( x ) ,  for all  x ∈ X . H(x) <= F(x),quad" for all "x in X.H(x) \leq F(x), \quad \text { for all } x \in X .H(x)≤F(x), for all x∈X.
Also, we have
H ( y ) − H ( x ) ≤ ‖ H ‖ X ⋅ d ( y , x ) = ‖ f ‖ Y ⋅ d ( y , x ) H ( y ) − H ( x ) ≤ ‖ H ‖ X ⋅ d ( y , x ) = ‖ f ‖ Y ⋅ d ( y , x ) 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)H(y)−H(x)≤‖H‖X⋅d(y,x)=‖f‖Y⋅d(y,x)
which implies
H ( x ) ≥ H ( y ) − ‖ f ‖ Y ⋅ d ( y , x ) = f ( y ) − ‖ f ‖ Y ⋅ d ( y , x ) H ( x ) ≥ H ( y ) − ‖ f ‖ Y ⋅ d ( y , x ) = f ( y ) − ‖ f ‖ Y ⋅ d ( y , x ) 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)H(x)≥H(y)−‖f‖Y⋅d(y,x)=f(y)−‖f‖Y⋅d(y,x)
Taking the supremum with respect to y ∈ Y y ∈ Y y in Yy \in Yy∈Y we get
H ( x ) ≥ G ( x ) , x ∈ X . H ( x ) ≥ G ( x ) , x ∈ X . H(x) >= G(x),quad x in X.H(x) \geq G(x), \quad x \in X .H(x)≥G(x),x∈X.
c) Let f ∈ ext B Y f ∈ ext B Y f in extB_(Y)f \in \operatorname{ext} B_{Y}f∈extBY. If F 1 , F 2 ∈ B X F 1 , F 2 ∈ B X F_(1),F_(2)inB_(X)F_{1}, F_{2} \in B_{X}F1,F2∈BX and λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)λ∈(0,1) are such that λ F 1 + ( 1 − λ ) F 2 ∈ E Y ( f ) λ F 1 + ( 1 − λ ) F 2 ∈ E Y ( f ) lambdaF_(1)+(1-lambda)F_(2)inE_(Y)(f)\lambda F_{1}+ (1-\lambda) F_{2} \in E_{Y}(f)λF1+(1−λ)F2∈EY(f) then λ F 1 | Y + ( 1 − λ ) F 2 | Y = f λ F 1 Y + ( 1 − λ ) F 2 Y = f lambdaF_(1)|_(Y)+(1-lambda)F_(2)|_(Y)=f\left.\lambda F_{1}\right|_{Y}+\left.(1-\lambda) F_{2}\right|_{Y}=fλF1|Y+(1−λ)F2|Y=f. Since f ∈ ext B Y f ∈ ext B Y f in extB_(Y)f \in \operatorname{ext} B_{Y}f∈extBY this implies F 1 | Y = F 2 | Y = f F 1 Y = F 2 Y = f F_(1)|_(Y)=F_(2)|_(Y)=f\left.F_{1}\right|_{Y}=\left.F_{2}\right|_{Y}=fF1|Y=F2|Y=f. Obviously that ‖ F 1 ‖ X = ‖ F 2 ‖ X = ‖ f ‖ Y = 1 F 1 X = F 2 X = ‖ f ‖ Y = 1 ||F_(1)||_(X)=||F_(2)||_(X)=||f||_(Y)=1\left\|F_{1}\right\|_{X}=\left\|F_{2}\right\|_{X}=\|f\|_{Y}=1‖F1‖X=‖F2‖X=‖f‖Y=1, showing that F 1 , F 2 ∈ E Y ( f ) F 1 , F 2 ∈ E Y ( f ) F_(1),F_(2)inE_(Y)(f)F_{1}, F_{2} \in E_{Y}(f)F1,F2∈EY(f). It follows that E Y ( f ) E Y ( f ) E_(Y)(f)E_{Y}(f)EY(f) is a face of B X B X B_(X)B_{X}BX.
We remark that F , G F , G F,GF, GF,G defined by (8), (9) are extremal elements of E Y ( f ) E Y ( f ) E_(Y)(f)E_{Y}(f)EY(f).
Indeed, if H 1 , H 2 ∈ E Y ( f ) H 1 , H 2 ∈ E Y ( f ) H_(1),H_(2)inE_(Y)(f)H_{1}, H_{2} \in E_{Y}(f)H1,H2∈EY(f) and λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)λ∈(0,1) are such that λ H 1 + ( 1 − λ ) H 2 = F λ H 1 + ( 1 − λ ) H 2 = F lambdaH_(1)+(1-lambda)H_(2)=F\lambda H_{1}+(1-\lambda) H_{2}= FλH1+(1−λ)H2=F then λ H 1 | Y + ( 1 − λ ) H 2 | Y = f λ H 1 Y + ( 1 − λ ) H 2 Y = f lambdaH_(1)|_(Y)+(1-lambda)H_(2)|_(Y)=f\left.\lambda H_{1}\right|_{Y}+\left.(1-\lambda) H_{2}\right|_{Y}=fλH1|Y+(1−λ)H2|Y=f and because f ∈ ext B Y f ∈ ext B Y f in extB_(Y)f \in \operatorname{ext} B_{Y}f∈extBY it follows H 1 | Y = H 2 | Y = f = F | Y H 1 Y = H 2 Y = f = F Y H_(1)|_(Y)=H_(2)|_(Y)=f=F|_(Y)\left.H_{1}\right|_{Y}= \left.H_{2}\right|_{Y}=f=\left.F\right|_{Y}H1|Y=H2|Y=f=F|Y.
On the other hand λ H 1 ( x ) + ( 1 − λ ) H 2 ( x ) = F ( x ) , x ∈ X λ H 1 ( x ) + ( 1 − λ ) H 2 ( x ) = F ( x ) , x ∈ X 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 XλH1(x)+(1−λ)H2(x)=F(x),x∈X implies
λ ( H 1 ( x ) − F ( x ) ) + ( 1 − λ ) ( H 2 ( x ) − F ( x ) ) = 0 , x ∈ X λ H 1 ( x ) − F ( x ) + ( 1 − λ ) H 2 ( x ) − F ( x ) = 0 , x ∈ X 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 Xλ(H1(x)−F(x))+(1−λ)(H2(x)−F(x))=0,x∈X
and because H 1 ( x ) ≤ F ( x ) , H 2 ( x ) ≤ F ( x ) , x ∈ X H 1 ( x ) ≤ F ( x ) , H 2 ( x ) ≤ F ( x ) , x ∈ X 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 XH1(x)≤F(x),H2(x)≤F(x),x∈X and λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)λ∈(0,1) it follows
H 1 ( x ) = F ( x ) , x ∈ X , H 2 ( x ) = F ( x ) , x ∈ X . H 1 ( x ) = F ( x ) ,      x ∈ X , H 2 ( x ) = F ( x ) ,      x ∈ X . {:[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}H1(x)=F(x),x∈X,H2(x)=F(x),x∈X.
Consequently F ∈ ext E Y ( f ) F ∈ ext E Y ( f ) F in extE_(Y)(f)F \in \operatorname{ext} E_{Y}(f)F∈extEY(f). Analogously one obtains G ∈ ext E Y ( f ) G ∈ ext E Y ( f ) G in extE_(Y)(f)G \in \operatorname{ext} E_{Y}(f)G∈extEY(f).
Now let be given U 1 , U 2 ∈ B X U 1 , U 2 ∈ B X U_(1),U_(2)inB_(X)U_{1}, U_{2} \in B_{X}U1,U2∈BX and λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)λ∈(0,1) such that λ U 1 + ( 1 − λ ) U 2 = F λ U 1 + ( 1 − λ ) U 2 = F lambdaU_(1)+(1-lambda)U_(2)=F\lambda U_{1}+(1-\lambda) U_{2}=FλU1+(1−λ)U2=F. Then λ U 1 | Y + ( 1 − λ ) U 2 | Y = F | Y = f ∈ ext B Y λ U 1 Y + ( 1 − λ ) U 2 Y = F Y = f ∈ ext B Y 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}λU1|Y+(1−λ)U2|Y=F|Y=f∈extBY implies U 1 | Y = U 2 | Y = f U 1 Y = U 2 Y = f U_(1)|_(Y)=U_(2)|_(Y)=f\left.U_{1}\right|_{Y}=\left.U_{2}\right|_{Y}=fU1|Y=U2|Y=f and ‖ U 1 | Y ‖ Y = ‖ U 2 | Y ‖ Y = ‖ f ‖ = 1 U 1 Y Y = U 2 Y Y = ‖ f ‖ = 1 ||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‖U1|Y‖Y=‖U2|Y‖Y=‖f‖=1 implies ‖ U 1 ‖ X = ‖ U 2 ‖ X = 1 U 1 X = U 2 X = 1 ||U_(1)||_(X)=||U_(2)||_(X)=1\left\|U_{1}\right\|_{X}=\left\|U_{2}\right\|_{X}=1‖U1‖X=‖U2‖X=1.
It follows that U 1 , U 2 ∈ E Y ( f ) U 1 , U 2 ∈ E Y ( f ) U_(1),U_(2)inE_(Y)(f)U_{1}, U_{2} \in E_{Y}(f)U1,U2∈EY(f) and because F ∈ ext E Y ( f ) F ∈ ext E Y ( f ) F in extE_(Y)(f)F \in \operatorname{ext} E_{Y}(f)F∈extEY(f) one obtains U 1 = U 2 = F U 1 = U 2 = F U_(1)=U_(2)=FU_{1}=U_{2}=FU1=U2=F. It follows that F ∈ ext B X F ∈ ext B X F in extB_(X)F \in \operatorname{ext} B_{X}F∈extBX and, analogously G ∈ ext B X G ∈ ext B X G in extB_(X)G \in \operatorname{ext} B_{X}G∈extBX.
Remarks. 1 ∘ 1 ∘ 1^(@)1^{\circ}1∘. The reverse implication in c) is also true: if E Y ( f ) E Y ( f ) E_(Y)(f)E_{Y}(f)EY(f) is a face of B X B X B_(X)B_{X}BX then f ∈ B Y f ∈ B Y f inB_(Y)f \in B_{Y}f∈BY.
Indeed, if ‖ f ‖ Y = 1 ‖ f ‖ Y = 1 ||f||_(Y)=1\|f\|_{Y}=1‖f‖Y=1 but f ∉ ext B Y f ∉ ext B Y f!in extB_(Y)f \notin \operatorname{ext} B_{Y}f∉extBY, then there exist f 1 , f 2 ∈ B Y , f 1 ≠ f 2 f 1 , f 2 ∈ B Y , f 1 ≠ f 2 f_(1),f_(2)inB_(Y),f_(1)!=f_(2)f_{1}, f_{2} \in B_{Y}, f_{1} \neq f_{2}f1,f2∈BY,f1≠f2, and λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)λ∈(0,1) such that λ f 1 + ( 1 − λ ) f 2 = f λ f 1 + ( 1 − λ ) f 2 = f lambdaf_(1)+(1-lambda)f_(2)=f\lambda f_{1}+(1-\lambda) f_{2}=fλf1+(1−λ)f2=f.
Let F 1 ′ ∈ E Y ( f 1 ) F 1 ′ ∈ E Y f 1 F_(1)^(')inE_(Y)(f_(1))F_{1}^{\prime} \in E_{Y}\left(f_{1}\right)F1′∈EY(f1) and F 2 ′ ∈ E Y ( f 2 ) F 2 ′ ∈ E Y f 2 F_(2)^(')inE_(Y)(f_(2))F_{2}^{\prime} \in E_{Y}\left(f_{2}\right)F2′∈EY(f2). Because
λ F 1 ′ | Y + ( 1 − λ ) F 2 ′ | Y = f λ F 1 ′ Y + ( 1 − λ ) F 2 ′ Y = f lambdaF_(1)^(')|_(Y)+(1-lambda)F_(2)^(')|_(Y)=f\left.\lambda F_{1}^{\prime}\right|_{Y}+\left.(1-\lambda) F_{2}^{\prime}\right|_{Y}=fλF1′|Y+(1−λ)F2′|Y=f
and
1 = ‖ λ F 1 ′ | Y + ( 1 − λ ) F 2 ′ | Y ‖ ≤ ‖ λ F 1 ′ + ( 1 − λ ) F 2 ′ ‖ X ≤ 1 1 = λ F 1 ′ Y + ( 1 − λ ) F 2 ′ Y ≤ λ F 1 ′ + ( 1 − λ ) F 2 ′ X ≤ 1 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 11=‖λF1′|Y+(1−λ)F2′|Y‖≤‖λF1′+(1−λ)F2′‖X≤1
we have
‖ λ F 1 ′ + ( 1 − λ ) F 2 ′ ‖ X = 1 λ F 1 ′ + ( 1 − λ ) F 2 ′ X = 1 ||lambdaF_(1)^(')+(1-lambda)F_(2)^(')||_(X)=1\left\|\lambda F_{1}^{\prime}+(1-\lambda) F_{2}^{\prime}\right\|_{X}=1‖λF1′+(1−λ)F2′‖X=1
showing that λ F 1 ′ + ( 1 − λ ) F 2 ′ ∈ E Y ( f ) λ F 1 ′ + ( 1 − λ ) F 2 ′ ∈ E Y ( f ) lambdaF_(1)^(')+(1-lambda)F_(2)^(')inE_(Y)(f)\lambda F_{1}^{\prime}+(1-\lambda) F_{2}^{\prime} \in E_{Y}(f)λF1′+(1−λ)F2′∈EY(f). Since F 1 ′ | Y = f 1 ≠ f 2 = F 2 ′ | Y F 1 ′ Y = f 1 ≠ f 2 = F 2 ′ Y 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}F1′|Y=f1≠f2=F2′|Y it follows that E Y ( f ) E Y ( f ) E_(Y)(f)E_{Y}(f)EY(f) is not a face of B X B X B_(X)B_{X}BX.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘. The assertion c) from Theorem 2 gives us a way to obtain extremal elements of B X B X B_(X)B_{X}BX, namely as the extensions (8) and (9) of extremal elements of B Y B Y B_(Y)B_{Y}BY.
Example. Consider the quasi metric space ( R , d ) ( R , d ) (R,d)(\mathbb{R}, d)(R,d), where R R R\mathbb{R}R is the set of real numbers and
d ( x , y ) = { x − y if x ≥ y 1 if x < y d ( x , y ) = x − y  if  x ≥ y 1  if  x < y d(x,y)={[x-y," if ",x >= y],[1," if ",x < y]:}d(x, y)=\left\{\begin{array}{ccc} x-y & \text { if } & x \geq y \\ 1 & \text { if } & x<y \end{array}\right.d(x,y)={x−y if x≥y1 if x<y
For Y = { 0 , 1 , 2 } Y = { 0 , 1 , 2 } Y={0,1,2}Y=\{0,1,2\}Y={0,1,2} and x 0 = 0 x 0 = 0 x_(0)=0x_{0}=0x0=0 consider the semilinear spaces SLip 0 Y SLip 0 Y SLip_(0)Y\operatorname{SLip}_{0} YSLip0Y and S L i p 0 X S L i p 0 X SLip_(0)XS L i p_{0} XSLip0X equipped with the quasi-norms of the type (3).
The function f ( y ) = y , y ∈ { 0 , 1 , 2 } = Y f ( y ) = y , y ∈ { 0 , 1 , 2 } = Y f(y)=y,y in{0,1,2}=Yf(y)=y, y \in\{0,1,2\}=Yf(y)=y,y∈{0,1,2}=Y is an extremal element of B Y B Y B_(Y)B_{Y}BY. Observe that for any h ∈ B Y h ∈ B Y h inB_(Y)h \in B_{Y}h∈BY we have h ( 1 ) ≤ f ( 1 ) = 1 h ( 1 ) ≤ f ( 1 ) = 1 h(1) <= f(1)=1h(1) \leq f(1)=1h(1)≤f(1)=1 and h ( 2 ) ≤ f ( 2 ) = 2 h ( 2 ) ≤ f ( 2 ) = 2 h(2) <= f(2)=2h(2) \leq f(2)=2h(2)≤f(2)=2, because, if contrary, i.e. h ( 1 ) > f ( 1 ) h ( 1 ) > f ( 1 ) h(1) > f(1)h(1)>f(1)h(1)>f(1) or h ( 2 ) > f ( 2 ) h ( 2 ) > f ( 2 ) h(2) > f(2)h(2)>f(2)h(2)>f(2), then ‖ h ‖ Y > 1 ‖ h ‖ Y > 1 ||h||_(Y) > 1\|h\|_{Y}>1‖h‖Y>1. If f 1 , f 2 ∈ B Y f 1 , f 2 ∈ B Y f_(1),f_(2)inB_(Y)f_{1}, f_{2} \in B_{Y}f1,f2∈BY and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) are such that α f 1 + ( 1 − α ) f 2 = f α f 1 + ( 1 − α ) f 2 = f alphaf_(1)+(1-alpha)f_(2)=f\alpha f_{1}+(1-\alpha) f_{2}=fαf1+(1−α)f2=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 , 2 f i ( 0 ) = f ( 0 ) = 0 , f i ( 1 ) ≤ f ( 1 ) , f i ( 2 ) ≤ f ( 2 ) , i = 1 , 2 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=1,2fi(0)=f(0)=0,fi(1)≤f(1),fi(2)≤f(2),i=1,2, we get α ( f 1 − f ) + ( 1 − α ) ( f 2 − f ) = 0 α f 1 − f + ( 1 − α ) f 2 − f = 0 alpha(f_(1)-f)+(1-alpha)(f_(2)-f)=0\alpha\left(f_{1}-f\right)+(1-\alpha)\left(f_{2}-f\right)=0α(f1−f)+(1−α)(f2−f)=0 implying f 1 = f 2 = f f 1 = f 2 = f f_(1)=f_(2)=ff_{1}=f_{2}=ff1=f2=f.
In this case the extensions F F FFF and G G GGG, given by (8) and (9), are
F ( x ) = { 1 , for x ∈ ( − ∞ , 0 ) x , for x ∈ [ 0 , + ∞ ) respectively G ( x ) = { x , for x ∈ ( − ∞ , 2 ] 1 , for x ∈ ( 2 , + ∞ ) F ( x ) = 1 ,       for  x ∈ ( − ∞ , 0 ) x ,       for  x ∈ [ 0 , + ∞ )  respectively  G ( x ) = x ,       for  x ∈ ( − ∞ , 2 ] 1 ,       for  x ∈ ( 2 , + ∞ ) 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.F(x)={1, for x∈(−∞,0)x, for x∈[0,+∞) respectively G(x)={x, for x∈(−∞,2]1, for x∈(2,+∞)
and they are extremal elements of B X B X B_(X)B_{X}BX.

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] Mustăţa, 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). ©
[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.
  1. *"T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, 3400 Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro.

Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.