Extension of convex semi-Lipschitz functions on quasi-metric linear spaces

Costica Mustata
(original), Approximation Theory, asymmetric norm, paper

Abstract

In this paper one shows that  a convex semi-Lipschitz functions defined on a convex subset of a quasi-metric linear spaces X admits an extension to the vohle spaces X, preserving both the convexity and the semi-Lipschitz constant. A similar result is proved  for starshaped  functions.

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

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C. Mustăţa, Extension of convex semi-Lipschitz Functions on quasi-metric linear spaces, Seminaire de la Théorie de la Meilleure Approximation, Convexité et Optimization, Cluj-Napoca, 29 November 2001, 85-92.

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[1] Romaguera, S. , Sanchis, M, Semi-Lipschitz functions  in quasi-metric spaces, J.A.T. 103, (2000), 292-301.
[2] McShane, J.A., Extension of range of functions , Bull. Amer. Math. Soc., 40 (1934), 837-842.
[3] Cobzas, S, Mustata, C., Norm preserving extension of convex Lipschitz functions, J.A.T. 24(1978), 555-564.
[4] Mustata, C., On the extension of semi-Lipschitz functions on quasi-metric space (to appear).
[5] Wels, J.H, Williams, L.R., Embeddings and extension in analysis, Springer-Verlag , Berlin, 1975.

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2001-Mustata-Seminar-Meilleure-Extensions-of-convex-semi-Lipschitz-functions-on-quasi-metric-linear-

Extensions of convex semi-Lipschitz functions on quasi-metric linear spaces

Costică Mustăţa(Cluj-Napoca)

Abstract

In this paper one shows that a convex semi-Lipschitz functions defined on a convex subset of a quasi-metric linear space X X XXX admits an extension to the whole space X X XXX, preserving both the convexity and the semi-Lipschitz constant. A similar result is proved for starshaped functions.

1 Introduction

Let X X XXX be a set. A function d : X × X → [ 0 , ∞ ) d : X × X → [ 0 , ∞ ) d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty)d:X×X→[0,∞) is called a quasi-metric if
(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),
for all x . y . z ∈ X x . y . z ∈ X x.y.z in Xx . y . z \in Xx.y.z∈X. If d d ddd is a quasi-metric on X X XXX then the pair ( X , d ) ( X , d ) (X,d)(X, d)(X,d) is called a quasi-metric space. If X X XXX is further a linear space and d d ddd is a quasi-metric on X X XXX then the pair ( X , d X , d X,dX, dX,d ) is called a quasi-metric linear space.
The function d − 1 : X × X → [ 0 , ∞ ) d − 1 : X × X → [ 0 , ∞ ) d^(-1):X xx X rarr[0,oo)d^{-1}: X \times X \rightarrow[0, \infty)d−1:X×X→[0,∞) defined by equality
(1) d − 1 ( x , y ) = d ( y , x ) , x , y ∈ X (1) d − 1 ( x , y ) = d ( y , x ) , x , y ∈ X {:(1)d^(-1)(x","y)=d(y","x)","quad x","y in X:}\begin{equation*} d^{-1}(x, y)=d(y, x), \quad x, y \in X \tag{1} \end{equation*}(1)d−1(x,y)=d(y,x),x,y∈X
is called the conjugate of the quasi-metric d [ 1 ] d [ 1 ] d[1]d[1]d[1].
Definition 1.1 Let ( X , d X , d X,dX, dX,d ) be a quasi-metric linear space and Y Y YYY a convex subset of X X XXX (i.e. ( ∀ ) u 1 , u 2 ∈ Y , ( ∀ ) α ∈ [ 0 , 1 ] , α u 1 + ( 1 − α ) u 2 ∈ Y ( ∀ ) u 1 , u 2 ∈ Y , ( ∀ ) α ∈ [ 0 , 1 ] , α u 1 + ( 1 − α ) u 2 ∈ Y (AA)u_(1),u_(2)in Y,(AA)alpha in[0,1],alphau_(1)+(1-alpha)u_(2)in Y(\forall) u_{1}, u_{2} \in Y,(\forall) \alpha \in[0,1], \alpha u_{1}+ (1-\alpha) u_{2} \in Y(∀)u1,u2∈Y,(∀)α∈[0,1],αu1+(1−α)u2∈Y ). The quasi-metric d d ddd is called convex on Y Y YYY if it satisfies the inequality
(2)
d ( α x 1 + ( 1 − α ) x 2 , α y 1 + ( 1 − α ) y 2 ) ≤ α d ( x 1 , y 1 ) + ( 1 − α ) d ( x 2 , y 2 ) d α x 1 + ( 1 − α ) x 2 , α y 1 + ( 1 − α ) y 2 ≤ α d x 1 , y 1 + ( 1 − α ) d x 2 , y 2 d(alphax_(1)+(1-alpha)x_(2),alphay_(1)+(1-alpha)y_(2)) <= alpha d(x_(1),y_(1))+(1-alpha)d(x_(2),y_(2))d\left(\alpha x_{1}+(1-\alpha) x_{2}, \alpha y_{1}+(1-\alpha) y_{2}\right) \leq \alpha d\left(x_{1}, y_{1}\right)+(1-\alpha) d\left(x_{2}, y_{2}\right)d(αx1+(1−α)x2,αy1+(1−α)y2)≤αd(x1,y1)+(1−α)d(x2,y2), for all x 1 , x 2 , y 1 , y 2 ∈ Y x 1 , x 2 , y 1 , y 2 ∈ Y x_(1),x_(2),y_(1),y_(2)in Yx_{1}, x_{2}, y_{1}, y_{2} \in Yx1,x2,y1,y2∈Y and all α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1].
Definition 1.2 [4]. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric linear space. A function f : X → R f : X → R f:X rarrRf: X \rightarrow \mathbb{R}f:X→R is called semi-Lipschitz if there exists K ≥ 0 K ≥ 0 K >= 0K \geq 0K≥0 such that
(3) f ( x ) − f ( y ) ≤ K ⋅ d ( x , y ) , (3) f ( x ) − f ( y ) ≤ K ⋅ d ( x , y ) , {:(3)f(x)-f(y) <= K*d(x","y)",":}\begin{equation*} f(x)-f(y) \leq K \cdot d(x, y), \tag{3} \end{equation*}(3)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 number K ≥ 0 K ≥ 0 K >= 0K \geq 0K≥0 for which (3) holds is called a semi-Lipschitz constant for f f fff.
For Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X let
(4) ‖ f ‖ Y = sup { ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) : x , y ∈ Y , d ( x , y ) > 0 } ‖ f ‖ Y = sup ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) : x , y ∈ Y , d ( x , y ) > 0 quad||f||_(Y)=s u p{((f(x)-f(y))vv0)/(d(x,y)):x,y in Y,d(x,y) > 0}\quad\|f\|_{Y}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: x, y \in Y, d(x, y)>0\right\}‖f‖Y=sup{(f(x)−f(y))∨0d(x,y):x,y∈Y,d(x,y)>0}.
The ‖ f ‖ Y ‖ f ‖ Y ||f||_(Y)\|f\|_{Y}‖f‖Y is the smallest semi-Lipschitz constant of the function f f fff on Y Y YYY [3, Th.1].
For Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X let
(5) S Lip Y = { f : Y → R : ‖ f ‖ Y < ∞ } (5) S Lip Y = f : Y → R : ‖ f ‖ Y < ∞ {:(5)S Lip Y={f:Y rarrR:||f||_(Y) < oo}:}\begin{equation*} S \operatorname{Lip} Y=\left\{f: Y \rightarrow \mathbb{R}:\|f\|_{Y}<\infty\right\} \tag{5} \end{equation*}(5)SLipY={f:Y→R:‖f‖Y<∞}
be the set of all real-valued semi-Lipschitz functions defined on the quasi-metric space ( Y , d ) ( Y , d ) (Y,d)(Y, d)(Y,d).
Definition 1.3 A function F ∈ S F ∈ S F in SF \in SF∈S Lip X X XXX is called an extension of a function f ∈ S f ∈ S f in Sf \in Sf∈S Lip Y Y YYY if
(i) F | Y = f F Y = f F|_(Y)=f\left.F\right|_{Y}=fF|Y=f,
(ii) ‖ F ‖ X = ‖ f ‖ Y ‖ F ‖ X = ‖ f ‖ Y ||F||_(X)=||f||_(Y)\|F\|_{X}=\|f\|_{Y}‖F‖X=‖f‖Y.
For f ∈ S Lip Y f ∈ S Lip Y f in S Lip Yf \in S \operatorname{Lip} Yf∈SLipY one denotes by
(6) E Y ( f ) = { F ∈ S Lip X : F | Y = f and ‖ F ‖ X = ‖ f ‖ Y } (6) E Y ( f ) = F ∈ S Lip X : F Y = f  and  ‖ F ‖ X = ‖ f ‖ Y {:(6)E_(Y)(f)={F in S Lip X:F|_(Y)=f" and "||F||_(X)=||f||_(Y)}:}\begin{equation*} E_{Y}(f)=\left\{F \in S \operatorname{Lip} X:\left.F\right|_{Y}=f \text { and }\|F\|_{X}=\|f\|_{Y}\right\} \tag{6} \end{equation*}(6)EY(f)={F∈SLipX:F|Y=f and ‖F‖X=‖f‖Y}
the set of all extensions of the function f f fff.
By Theorem 2 in [4] it follows that
(7) E Y ( f ) ≠ ∅ , ( ∀ ) f ∈ S Lip Y . (7) E Y ( f ) ≠ ∅ , ( ∀ ) f ∈ S Lip Y . {:(7)E_(Y)(f)!=O/","(AA)f in S Lip Y.:}\begin{equation*} E_{Y}(f) \neq \emptyset,(\forall) f \in S \operatorname{Lip} Y . \tag{7} \end{equation*}(7)EY(f)≠∅,(∀)f∈SLipY.
The following problem arises naturally: which other properties of the function f f fff (beside the semi-Lipschitz constant) are preserved by at least one of its extensions?
The aim of this paper is to show that two such properties are convexity and starshapedness.
First we prove:
Theorem 1.1 Let ( X , d X , d X,dX, dX,d ) be a quasi-metric linear space and Y Y YYY a convex subset of X X XXX. Suppose that the quasi-metric d d ddd is convex on Y Y YYY, in the sense of Definition 1.
a) If f ∈ S f ∈ S f in Sf \in Sf∈S Lip Y Y YYY is convex on Y Y YYY then there exists a convex
F ∈ E Y ( f ) . F ∈ E Y ( f ) . F inE_(Y)(f).F \in E_{Y}(f) .F∈EY(f).
b) If f ∈ S f ∈ S f in Sf \in Sf∈S Lip Y Y YYY is concave (i.e. − f − f -f-f−f is convex) on Y Y YYY then there exists a concave
G ∈ E Y ( f ) . G ∈ E Y ( f ) . G inE_(Y)(f).G \in E_{Y}(f) .G∈EY(f).
Proof. a) Let f ∈ S Lip Y f ∈ S Lip Y f in S Lip Yf \in S \operatorname{Lip} Yf∈SLipY be convex on the convex set Y Y YYY. Consider the function F : X → R F : X → R F:X rarrRF: X \rightarrow \mathbb{R}F:X→R defined by
(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)}","x in X.:}\begin{equation*} F(x)=\inf _{y \in Y}\left\{f(y)+\|f\|_{Y} \cdot d(x, y)\right\}, x \in X . \tag{8} \end{equation*}(8)F(x)=infy∈Y{f(y)+‖f‖Y⋅d(x,y)},x∈X.
Then F F FFF is well defined and F ∈ E Y ( f ) F ∈ E Y ( f ) F inE_(Y)(f)F \in E_{Y}(f)F∈EY(f) (Theorem 2 in [4]). Indeed, let z ∈ Y z ∈ Y z in Yz \in Yz∈Y and x ∈ X x ∈ X x in Xx \in Xx∈X. For any y ∈ Y y ∈ Y y in Yy \in Yy∈Y we have
f ( y ) + ‖ f ‖ Y d ( x , y ) = f ( z ) + ‖ f ‖ Y d ( x , y ) − ( f ( z ) − f ( y ) ) ≥ f ( z ) + ‖ f ‖ Y d ( x , y ) − ‖ f ‖ Y d ( z , y ) = f ( z ) − ‖ f ‖ Y ( d ( z , y ) − d ( x , y ) ) . f ( y ) + ‖ f ‖ Y d ( x , y ) = f ( z ) + ‖ f ‖ Y d ( x , y ) − ( f ( z ) − f ( y ) ) ≥ f ( z ) + ‖ f ‖ Y d ( x , y ) − ‖ f ‖ Y d ( z , y ) = f ( z ) − ‖ f ‖ Y ( d ( z , y ) − d ( x , y ) ) . {:[f(y)+||f||_(Y)d(x","y)=f(z)+||f||_(Y)d(x","y)-(f(z)-f(y))],[ >= f(z)+||f||_(Y)d(x","y)-||f||_(Y)d(z","y)],[=f(z)-||f||_(Y)(d(z","y)-d(x","y)).]:}\begin{aligned} f(y)+\|f\|_{Y} d(x, y) & =f(z)+\|f\|_{Y} d(x, y)-(f(z)-f(y)) \\ & \geq f(z)+\|f\|_{Y} d(x, y)-\|f\|_{Y} d(z, y) \\ & =f(z)-\|f\|_{Y}(d(z, y)-d(x, y)) . \end{aligned}f(y)+‖f‖Yd(x,y)=f(z)+‖f‖Yd(x,y)−(f(z)−f(y))≥f(z)+‖f‖Yd(x,y)−‖f‖Yd(z,y)=f(z)−‖f‖Y(d(z,y)−d(x,y)).
The inequality d ( z , y ) − d ( x , y ) ≤ d ( z , x ) = d − 1 ( x , z ) d ( z , y ) − d ( x , y ) ≤ d ( z , x ) = d − 1 ( x , z ) d(z,y)-d(x,y) <= d(z,x)=d^(-1)(x,z)d(z, y)-d(x, y) \leq d(z, x)=d^{-1}(x, z)d(z,y)−d(x,y)≤d(z,x)=d−1(x,z) implies
f ( y ) + ‖ f ‖ Y d ( x , y ) ≥ f ( z ) − ‖ f ‖ Y ⋅ d − 1 ( x , z ) f ( y ) + ‖ f ‖ Y d ( x , y ) ≥ f ( z ) − ‖ f ‖ Y ⋅ d − 1 ( x , z ) f(y)+||f||_(Y)d(x,y) >= f(z)-||f||_(Y)*d^(-1)(x,z)f(y)+\|f\|_{Y} d(x, y) \geq f(z)-\|f\|_{Y} \cdot d^{-1}(x, z)f(y)+‖f‖Yd(x,y)≥f(z)−‖f‖Y⋅d−1(x,z)
showing that for every x ∈ X x ∈ X x in Xx \in Xx∈X the set { f ( y ) + ‖ f ‖ Y d ( x , y ) : y ∈ Y } f ( y ) + ‖ f ‖ Y d ( x , y ) : y ∈ Y {f(y)+||f||_(Y)d(x,y):y in Y}\left\{f(y)+\|f\|_{Y} d(x, y): y \in Y\right\}{f(y)+‖f‖Yd(x,y):y∈Y} is bounded from above by f ( z ) − ‖ f ‖ Y d − 1 ( x , z ) f ( z ) − ‖ f ‖ Y d − 1 ( x , z ) f(z)-||f||_(Y)d^(-1)(x,z)f(z)-\|f\|_{Y} d^{-1}(x, z)f(z)−‖f‖Yd−1(x,z), and the infimum (8) is finite.
We show now that F ( y ) = f ( y ) F ( y ) = f ( y ) F(y)=f(y)F(y)=f(y)F(y)=f(y) for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y.
Let y ∈ Y y ∈ Y y in Yy \in Yy∈Y. Then
F ( y ) ≤ f ( y ) + ‖ f ‖ Y d ( y , y ) = f ( y ) . F ( y ) ≤ f ( y ) + ‖ f ‖ Y d ( y , y ) = f ( y ) . F(y) <= f(y)+||f||_(Y)d(y,y)=f(y).F(y) \leq f(y)+\|f\|_{Y} d(y, y)=f(y) .F(y)≤f(y)+‖f‖Yd(y,y)=f(y).
For any v ∈ Y v ∈ Y v in Yv \in Yv∈Y we have
f ( y ) − f ( v ) ≤ ‖ f ‖ Y ⋅ d ( y , v ) f ( y ) − f ( v ) ≤ ‖ f ‖ Y ⋅ d ( y , v ) f(y)-f(v) <= ||f||_(Y)*d(y,v)f(y)-f(v) \leq\|f\|_{Y} \cdot d(y, v)f(y)−f(v)≤‖f‖Y⋅d(y,v)
so that
f ( v ) + ‖ f ‖ Y ⋅ d ( y , v ) ≥ f ( y ) f ( v ) + ‖ f ‖ Y ⋅ d ( y , v ) ≥ f ( y ) f(v)+||f||_(Y)*d(y,v) >= f(y)f(v)+\|f\|_{Y} \cdot d(y, v) \geq f(y)f(v)+‖f‖Y⋅d(y,v)≥f(y)
and
F ( y ) = inf { f ( v ) + ‖ f ‖ Y d ( y , v ) : v ∈ Y } ≥ f ( y ) F ( y ) = inf f ( v ) + ‖ f ‖ Y d ( y , v ) : v ∈ Y ≥ f ( y ) F(y)=i n f{f(v)+||f||_(Y)d(y,v):v in Y} >= f(y)F(y)=\inf \left\{f(v)+\|f\|_{Y} d(y, v): v \in Y\right\} \geq f(y)F(y)=inf{f(v)+‖f‖Yd(y,v):v∈Y}≥f(y)
It follows F ( y ) = f ( y ) F ( y ) = f ( y ) F(y)=f(y)F(y)=f(y)F(y)=f(y).
We prove that ‖ F ‖ Y = ‖ f ‖ Y ‖ F ‖ Y = ‖ f ‖ Y ||F||_(Y)=||f||_(Y)\|F\|_{Y}=\|f\|_{Y}‖F‖Y=‖f‖Y.
Since F | Y = f F Y = f F|_(Y)=f\left.F\right|_{Y}=fF|Y=f, the definitions of ‖ F ‖ Y ‖ F ‖ Y ||F||_(Y)\|F\|_{Y}‖F‖Y and ‖ f ‖ Y ‖ f ‖ Y ||f||_(Y)\|f\|_{Y}‖f‖Y yield ‖ F ‖ Y ≥ ‖ f ‖ Y ‖ F ‖ Y ≥ ‖ f ‖ Y ||F||_(Y) >= ||f||_(Y)\|F\|_{Y} \geq \|f\|_{Y}‖F‖Y≥‖f‖Y.
Let x 1 , x 2 ∈ X x 1 , x 2 ∈ X x_(1),x_(2)in Xx_{1}, x_{2} \in Xx1,x2∈X and ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0. Choosing y ∈ Y y ∈ Y y in Yy \in Yy∈Y such that
F ( x 1 ) ≥ f ( y ) + ‖ f ‖ Y d ( x 1 , y ) − ε F x 1 ≥ f ( y ) + ‖ f ‖ Y d x 1 , y − ε F(x_(1)) >= f(y)+||f||_(Y)d(x_(1),y)-epsiF\left(x_{1}\right) \geq f(y)+\|f\|_{Y} d\left(x_{1}, y\right)-\varepsilonF(x1)≥f(y)+‖f‖Yd(x1,y)−ε
we obtain
F ( x 2 ) − F ( x 1 ) ≤ f ( y ) + ‖ f ‖ Y d ( x 2 , y ) − ( f ( y ) + ‖ f ‖ Y ⋅ d ( x 1 , y ) − ε ) = ‖ f ‖ Y [ d ( x 2 , y ) − d ( x 1 , y ) ] + ε ≤ ‖ f ‖ Y ⋅ d ( x 2 , x 1 ) + ε . F x 2 − F x 1 ≤ f ( y ) + ‖ f ‖ Y d x 2 , y − f ( y ) + ‖ f ‖ Y ⋅ d x 1 , y − ε = ‖ f ‖ Y d x 2 , y − d x 1 , y + ε ≤ ‖ f ‖ Y ⋅ d x 2 , x 1 + ε . {:[F(x_(2))-F(x_(1))],[ <= f(y)+||f||_(Y)d(x_(2),y)-(f(y)+||f||_(Y)*d(x_(1),y)-epsi)],[=||f||_(Y)[d(x_(2),y)-d(x_(1),y)]+epsi],[ <= ||f||_(Y)*d(x_(2),x_(1))+epsi.]:}\begin{aligned} & F\left(x_{2}\right)-F\left(x_{1}\right) \\ \leq & f(y)+\|f\|_{Y} d\left(x_{2}, y\right)-\left(f(y)+\|f\|_{Y} \cdot d\left(x_{1}, y\right)-\varepsilon\right) \\ = & \|f\|_{Y}\left[d\left(x_{2}, y\right)-d\left(x_{1}, y\right)\right]+\varepsilon \\ \leq & \|f\|_{Y} \cdot d\left(x_{2}, x_{1}\right)+\varepsilon . \end{aligned}F(x2)−F(x1)≤f(y)+‖f‖Yd(x2,y)−(f(y)+‖f‖Y⋅d(x1,y)−ε)=‖f‖Y[d(x2,y)−d(x1,y)]+ε≤‖f‖Y⋅d(x2,x1)+ε.
Since ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 is arbitrary it follows
F ( x 2 ) − F ( x 1 ) ≤ ‖ f ‖ Y ⋅ d ( x 2 , x 1 ) F x 2 − F x 1 ≤ ‖ f ‖ Y ⋅ d x 2 , x 1 F(x_(2))-F(x_(1)) <= ||f||_(Y)*d(x_(2),x_(1))F\left(x_{2}\right)-F\left(x_{1}\right) \leq\|f\|_{Y} \cdot d\left(x_{2}, x_{1}\right)F(x2)−F(x1)≤‖f‖Y⋅d(x2,x1)
for any x 1 , x 2 ∈ X x 1 , x 2 ∈ X x_(1),x_(2)in Xx_{1}, x_{2} \in Xx1,x2∈X and ‖ F ‖ Y ≤ ‖ f ‖ Y ‖ F ‖ Y ≤ ‖ f ‖ Y ||F||_(Y) <= ||f||_(Y)\|F\|_{Y} \leq\|f\|_{Y}‖F‖Y≤‖f‖Y. Then ‖ F ‖ X = ‖ f ‖ Y ‖ F ‖ X = ‖ f ‖ Y ||F||_(X)=||f||_(Y)\|F\|_{X}=\|f\|_{Y}‖F‖X=‖f‖Y.
Since Y Y YYY is convex and the quasi-metric d d ddd is convex (in the sense
of Definition 1) we have
F ( α x 1 + ( 1 − α ) x 2 ) ≤ f ( α y 1 + ( 1 − α ) y 2 ) + + ‖ f ‖ Y ⋅ d ( α x 1 + ( 1 − α ) x 2 , α y 1 + ( 1 − α ) y 2 ) ≤ α f ( y 1 ) + ( 1 − α ) f ( y 2 ) + + ‖ f ‖ Y [ α d ( x 1 , y 1 ) + ( 1 − α ) d ( x 2 , y 2 ) ] = α [ f ( y 1 ) + ‖ f ‖ Y d ( x 1 , y 1 ) ] + + ( 1 − α ) [ f ( y 2 ) + ( 1 − α ) d ( x 2 , y 2 ) ] , F α x 1 + ( 1 − α ) x 2 ≤ f α y 1 + ( 1 − α ) y 2 + + ‖ f ‖ Y ⋅ d α x 1 + ( 1 − α ) x 2 , α y 1 + ( 1 − α ) y 2 ≤ α f y 1 + ( 1 − α ) f y 2 + + ‖ f ‖ Y α d x 1 , y 1 + ( 1 − α ) d x 2 , y 2 = α f y 1 + ‖ f ‖ Y d x 1 , y 1 + + ( 1 − α ) f y 2 + ( 1 − α ) d x 2 , y 2 , {:[F(alphax_(1)+(1-alpha)x_(2))],[ <= f(alphay_(1)+(1-alpha)y_(2))+],[+||f||_(Y)*d(alphax_(1)+(1-alpha)x_(2),alphay_(1)+(1-alpha)y_(2))],[ <= alpha f(y_(1))+(1-alpha)f(y_(2))+],[+||f||_(Y)[alpha d(x_(1),y_(1))+(1-alpha)d(x_(2),y_(2))]],[=alpha[f(y_(1))+||f||_(Y)d(x_(1),y_(1))]+],[+(1-alpha)[f(y_(2))+(1-alpha)d(x_(2),y_(2))]","]:}\begin{aligned} & F\left(\alpha x_{1}+(1-\alpha) x_{2}\right) \\ \leq & f\left(\alpha y_{1}+(1-\alpha) y_{2}\right)+ \\ & +\|f\|_{Y} \cdot d\left(\alpha x_{1}+(1-\alpha) x_{2}, \alpha y_{1}+(1-\alpha) y_{2}\right) \\ \leq & \alpha f\left(y_{1}\right)+(1-\alpha) f\left(y_{2}\right)+ \\ & +\|f\|_{Y}\left[\alpha d\left(x_{1}, y_{1}\right)+(1-\alpha) d\left(x_{2}, y_{2}\right)\right] \\ = & \alpha\left[f\left(y_{1}\right)+\|f\|_{Y} d\left(x_{1}, y_{1}\right)\right]+ \\ & +(1-\alpha)\left[f\left(y_{2}\right)+(1-\alpha) d\left(x_{2}, y_{2}\right)\right], \end{aligned}F(αx1+(1−α)x2)≤f(αy1+(1−α)y2)++‖f‖Y⋅d(αx1+(1−α)x2,αy1+(1−α)y2)≤αf(y1)+(1−α)f(y2)++‖f‖Y[αd(x1,y1)+(1−α)d(x2,y2)]=α[f(y1)+‖f‖Yd(x1,y1)]++(1−α)[f(y2)+(1−α)d(x2,y2)],
for all x 1 , x 2 ∈ X , y 1 , y 2 ∈ Y x 1 , x 2 ∈ X , y 1 , y 2 ∈ Y x_(1),x_(2)in X,y_(1),y_(2)in Yx_{1}, x_{2} \in X, y_{1}, y_{2} \in Yx1,x2∈X,y1,y2∈Y and α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1].
Taking the infimum with respect to y 1 , y 2 ∈ Y y 1 , y 2 ∈ Y y_(1),y_(2)in Yy_{1}, y_{2} \in Yy1,y2∈Y we obtain
(9) F ( α x 1 + ( 1 − α ) x 2 ) ≤ α F ( x 1 ) + ( 1 − α ) F ( x 2 ) (9) F α x 1 + ( 1 − α ) x 2 ≤ α F x 1 + ( 1 − α ) F x 2 {:(9)F(alphax_(1)+(1-alpha)x_(2)) <= alpha F(x_(1))+(1-alpha)F(x_(2)):}\begin{equation*} F\left(\alpha x_{1}+(1-\alpha) x_{2}\right) \leq \alpha F\left(x_{1}\right)+(1-\alpha) F\left(x_{2}\right) \tag{9} \end{equation*}(9)F(αx1+(1−α)x2)≤αF(x1)+(1−α)F(x2)
for all x 1 , x 2 ∈ X x 1 , x 2 ∈ X x_(1),x_(2)in Xx_{1}, x_{2} \in Xx1,x2∈X and all α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1], showing that the function F F FFF in E Y ( f ) E Y ( f ) E_(Y)(f)E_{Y}(f)EY(f), defined by (8), is convex.
b) If f ∈ S f ∈ S f in Sf \in Sf∈S Lip Y Y YYY is concave on Y Y YYY, let G : X → R G : X → R G:X rarrRG: X \rightarrow \mathbb{R}G:X→R be defined by
(10) G ( x ) = sup y ∈ Y { f ( y ) − ‖ f ‖ Y ⋅ d ( y , x ) } , x ∈ X (10) G ( x ) = sup y ∈ Y   f ( y ) − ‖ f ‖ Y ⋅ d ( y , x ) , x ∈ X {:(10)G(x)=s u p_(y in Y){f(y)-||f||_(Y)*d(y,x)}","x in X:}\begin{equation*} G(x)=\sup _{y \in Y}\left\{f(y)-\|f\|_{Y} \cdot d(y, x)\right\}, x \in X \tag{10} \end{equation*}(10)G(x)=supy∈Y{f(y)−‖f‖Y⋅d(y,x)},x∈X
Then G G GGG is well defined and G ∈ E Y ( f ) G ∈ E Y ( f ) G inE_(Y)(f)G \in E_{Y}(f)G∈EY(f) [4, Theorem 2].
For any x 1 , x 2 ∈ X , y 1 , y 2 ∈ Y x 1 , x 2 ∈ X , y 1 , y 2 ∈ Y x_(1),x_(2)in X,y_(1),y_(2)in Yx_{1}, x_{2} \in X, y_{1}, y_{2} \in Yx1,x2∈X,y1,y2∈Y and α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1] we have
G ( α x 1 + ( 1 − α ) x 2 ) ≥ f ( α y 1 + ( 1 − α ) y 2 ) − − ‖ f ‖ Y d ( α y 1 + ( 1 − α ) y 2 , α x 1 + ( 1 − α ) x 2 ) ≥ α f ( y 1 ) + ( 1 − α ) f ( y 2 ) − − ‖ f ‖ Y [ α ⋅ d ( y 1 , x 1 ) + ( 1 − α ) d ( y 2 , x 2 ) ] = α [ f ( y 1 ) − ‖ f ‖ Y d ( y 1 , x 1 ) ] + + ( 1 − α ) [ f ( y 2 ) − ‖ f ‖ Y d ( y 2 , x 2 ) ] G α x 1 + ( 1 − α ) x 2 ≥ f α y 1 + ( 1 − α ) y 2 − − ‖ f ‖ Y d α y 1 + ( 1 − α ) y 2 , α x 1 + ( 1 − α ) x 2 ≥ α f y 1 + ( 1 − α ) f y 2 − − ‖ f ‖ Y α ⋅ d y 1 , x 1 + ( 1 − α ) d y 2 , x 2 = α f y 1 − ‖ f ‖ Y d y 1 , x 1 + + ( 1 − α ) f y 2 − ‖ f ‖ Y d y 2 , x 2 {:[G(alphax_(1)+(1-alpha)x_(2))],[ >= f(alphay_(1)+(1-alpha)y_(2))-],[-||f||_(Y)d(alphay_(1)+(1-alpha)y_(2),alphax_(1)+(1-alpha)x_(2))],[ >= alpha f(y_(1))+(1-alpha)f(y_(2))-],[-||f||_(Y)[alpha*d(y_(1),x_(1))+(1-alpha)d(y_(2),x_(2))]],[=alpha[f(y_(1))-||f||_(Y)d(y_(1),x_(1))]+],[+(1-alpha)[f(y_(2))-||f||_(Y)d(y_(2),x_(2))]]:}\begin{aligned} & G\left(\alpha x_{1}+(1-\alpha) x_{2}\right) \\ \geq & f\left(\alpha y_{1}+(1-\alpha) y_{2}\right)- \\ & -\|f\|_{Y} d\left(\alpha y_{1}+(1-\alpha) y_{2}, \alpha x_{1}+(1-\alpha) x_{2}\right) \\ \geq & \alpha f\left(y_{1}\right)+(1-\alpha) f\left(y_{2}\right)- \\ & -\|f\|_{Y}\left[\alpha \cdot d\left(y_{1}, x_{1}\right)+(1-\alpha) d\left(y_{2}, x_{2}\right)\right] \\ = & \alpha\left[f\left(y_{1}\right)-\|f\|_{Y} d\left(y_{1}, x_{1}\right)\right]+ \\ & +(1-\alpha)\left[f\left(y_{2}\right)-\|f\|_{Y} d\left(y_{2}, x_{2}\right)\right] \end{aligned}G(αx1+(1−α)x2)≥f(αy1+(1−α)y2)−−‖f‖Yd(αy1+(1−α)y2,αx1+(1−α)x2)≥αf(y1)+(1−α)f(y2)−−‖f‖Y[α⋅d(y1,x1)+(1−α)d(y2,x2)]=α[f(y1)−‖f‖Yd(y1,x1)]++(1−α)[f(y2)−‖f‖Yd(y2,x2)]
Taking the supremum with respect to y 1 , y 2 ∈ Y y 1 , y 2 ∈ Y y_(1),y_(2)in Yy_{1}, y_{2} \in Yy1,y2∈Y we get
(11) G ( α x 1 + ( 1 − α ) x 2 ) ≥ α G ( x 1 ) + ( 1 − α ) G ( x 2 ) (11) G α x 1 + ( 1 − α ) x 2 ≥ α G x 1 + ( 1 − α ) G x 2 {:(11)G(alphax_(1)+(1-alpha)x_(2)) >= alpha G(x_(1))+(1-alpha)G(x_(2)):}\begin{equation*} G\left(\alpha x_{1}+(1-\alpha) x_{2}\right) \geq \alpha G\left(x_{1}\right)+(1-\alpha) G\left(x_{2}\right) \tag{11} \end{equation*}(11)G(αx1+(1−α)x2)≥αG(x1)+(1−α)G(x2)
for all x 1 , x 2 ∈ X x 1 , x 2 ∈ X x_(1),x_(2)in Xx_{1}, x_{2} \in Xx1,x2∈X and α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1], showing that the function G G GGG from E Y ( f ) E Y ( f ) E_(Y)(f)E_{Y}(f)EY(f), defined by (10), is concave.
Definition 1.4 Let X X XXX be a real linear space and θ ∈ Z ⊂ X θ ∈ Z ⊂ X theta in Z sub X\theta \in Z \subset Xθ∈Z⊂X where θ θ theta\thetaθ denotes the null element of X X XXX. The set Z Z ZZZ is called starshaped if
(12) ( ∀ ) α , α ∈ [ 0 , 1 ] , ( ∀ ) z ∈ Z , α z ∈ Z . (12) ( ∀ ) α , α ∈ [ 0 , 1 ] , ( ∀ ) z ∈ Z , α z ∈ Z . {:(12)(AA)alpha","alpha in[0","1]","(AA)z in Z","alpha z in Z.:}\begin{equation*} (\forall) \alpha, \alpha \in[0,1],(\forall) z \in Z, \alpha z \in Z . \tag{12} \end{equation*}(12)(∀)α,α∈[0,1],(∀)z∈Z,αz∈Z.
Obviously that any convex subset of X X XXX which contains θ θ theta\thetaθ is starshaped, and the converse is not true in general.
A function f : Z → R f : Z → R f:Z rarrRf: Z \rightarrow \mathbb{R}f:Z→R, where Z Z ZZZ is a starshaped subset of a linear space X X XXX, is called starshaped if
(12a) f ( α z ) ≤ α f ( z ) . (12a) f ( α z ) ≤ α f ( z ) . {:(12a)f(alpha z) <= alpha f(z).:}\begin{equation*} f(\alpha z) \leq \alpha f(z) . \tag{12a} \end{equation*}(12a)f(αz)≤αf(z).
for all z ∈ Z z ∈ Z z in Zz \in Zz∈Z and α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1]. A convex function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R, defined on a convex subset Y Y YYY of X X XXX containing θ θ theta\thetaθ, and such that f ( θ ) ≤ 0 f ( θ ) ≤ 0 f(theta) <= 0f(\theta) \leq 0f(θ)≤0 is starshaped but there are starshaped functions on Y Y YYY which are not convex.
Definition 1.5 Let ( X , d X , d X,dX, dX,d ) be a quasi-metric linear space. The quasi-metric d d ddd is called starshaped on X X XXX if the inequality
(13) d ( α x , α y ) ≤ α d ( x , y ) (13) d ( α x , α y ) ≤ α d ( x , y ) {:(13)d(alpha x","alpha y) <= alpha d(x","y):}\begin{equation*} d(\alpha x, \alpha y) \leq \alpha d(x, y) \tag{13} \end{equation*}(13)d(αx,αy)≤αd(x,y)
holds for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X and α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1].
Remark 1.1 If ( X , d ) ( X , d ) (X,d)(X, d)(X,d) is a quasi-metric linear space with convex quasi-metric, then d d ddd is starshaped because the inequality
d ( α x + ( 1 − α ) θ ) , α y + ( 1 − α ) θ ≤ α d ( x , y ) + ( 1 − α ) d ( θ , θ ) d ( α x + ( 1 − α ) θ ) , α y + ( 1 − α ) θ ≤ α d ( x , y ) + ( 1 − α ) d ( θ , θ ) d(alpha x+(1-alpha)theta),alpha y+(1-alpha)theta <= alpha d(x,y)+(1-alpha)d(theta,theta)d(\alpha x+(1-\alpha) \theta), \alpha y+(1-\alpha) \theta \leq \alpha d(x, y)+(1-\alpha) d(\theta, \theta)d(αx+(1−α)θ),αy+(1−α)θ≤αd(x,y)+(1−α)d(θ,θ)
yields
d ( α x , α y ) ≤ α d ( x , y ) d ( α x , α y ) ≤ α d ( x , y ) d(alpha x,alpha y) <= alpha d(x,y)d(\alpha x, \alpha y) \leq \alpha d(x, y)d(αx,αy)≤αd(x,y)
for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X and α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1].
Now we shall prove the extension result for starshaped semiLipschitz functions.
Theorem 1.2 Let ( X , d X , d X,dX, dX,d ) be a quasi-metric linear space with starshaped quasi-metric d d ddd and let Z Z ZZZ be a starshaped subset of X X XXX.
Then every starshaped function φ ∈ S φ ∈ S varphi in S\varphi \in Sφ∈S Lip Z Z ZZZ admits at least one starshaped extension Φ ∈ E Z ( φ ) Φ ∈ E Z ( φ ) Phi inE_(Z)(varphi)\Phi \in E_{Z}(\varphi)Φ∈EZ(φ).
Proof. Let φ ∈ S φ ∈ S varphi in S\varphi \in Sφ∈S Lip Z Z ZZZ be starshaped on the starshaped set Z ⊂ X Z ⊂ X Z sub XZ \subset XZ⊂X. The function
(14) Φ ( x ) = inf z ∈ Z { φ ( z ) + ‖ φ ‖ Z d ( x , z ) } , x ∈ X (14) Φ ( x ) = inf z ∈ Z   φ ( z ) + ‖ φ ‖ Z d ( x , z ) , x ∈ X {:(14)Phi(x)=i n f_(z in Z){varphi(z)+||varphi||_(Z)d(x,z)}","x in X:}\begin{equation*} \Phi(x)=\inf _{z \in Z}\left\{\varphi(z)+\|\varphi\|_{Z} d(x, z)\right\}, x \in X \tag{14} \end{equation*}(14)Φ(x)=infz∈Z{φ(z)+‖φ‖Zd(x,z)},x∈X
belongs to E Z ( φ ) E Z ( φ ) E_(Z)(varphi)E_{Z}(\varphi)EZ(φ) (Th 2 in [4]).
We shall show that Φ Φ Phi\PhiΦ is also starshaped on X X XXX. To this end let x ∈ X , z ∈ Z x ∈ X , z ∈ Z x in X,z in Zx \in X, z \in Zx∈X,z∈Z and α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1]. We have
Φ ( α x ) = φ ( α z ) + ‖ φ ‖ Z d ( α x , α z ) ≤ α φ ( z ) + α ‖ φ ‖ Z d ( x , z ) = α [ φ ( z ) + ‖ φ ‖ Z ⋅ d ( x , z ) ] . Φ ( α x ) = φ ( α z ) + ‖ φ ‖ Z d ( α x , α z ) ≤ α φ ( z ) + α ‖ φ ‖ Z d ( x , z ) = α φ ( z ) + ‖ φ ‖ Z ⋅ d ( x , z ) . {:[Phi(alpha x)=varphi(alpha z)+||varphi||_(Z)d(alpha x","alpha z)],[ <= alpha varphi(z)+alpha||varphi||_(Z)d(x","z)],[=alpha[varphi(z)+||varphi||_(Z)*d(x,z)].]:}\begin{aligned} \Phi(\alpha x) & =\varphi(\alpha z)+\|\varphi\|_{Z} d(\alpha x, \alpha z) \\ & \leq \alpha \varphi(z)+\alpha\|\varphi\|_{Z} d(x, z) \\ & =\alpha\left[\varphi(z)+\|\varphi\|_{Z} \cdot d(x, z)\right] . \end{aligned}Φ(αx)=φ(αz)+‖φ‖Zd(αx,αz)≤αφ(z)+α‖φ‖Zd(x,z)=α[φ(z)+‖φ‖Z⋅d(x,z)].
Taking the infimum with respect to z ∈ Z z ∈ Z z in Zz \in Zz∈Z we get
Φ ( α x ) ≤ α Φ ( x ) Φ ( α x ) ≤ α Φ ( x ) Phi(alpha x) <= alpha Phi(x)\Phi(\alpha x) \leq \alpha \Phi(x)Φ(αx)≤αΦ(x)
for all x ∈ X x ∈ X x in Xx \in Xx∈X and all α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1], showing that the function Φ Φ Phi\PhiΦ defined by (14) is a starshaped extension of φ φ varphi\varphiφ.

References

[1] Romaguera, S., Sanchis, M., Semi-Lipschitz Functions in Quasi-Metric Spaces, J.A.T 103 (2000), 292-301.
[2] McShane, J.A. Extension of range of functions, Bull.Amer.Math.Soc. 40 (1934), 837-842.
[3] Cobzaş, S., Mustăţa, C., Norm Preserving Extension of Convex Lipschitz Functions J.A.T. 24(1978) 555-564.
[4] Mustăta, C., On the Extension of Semi-Lipschitz Functions on Quasi-Metric space (to appear).
[5] Wels, J.H., Williams, L.R., Embeddings and Extension in Analysis, Springer-Verlag Berlin, 1975.
"T. Popoviciu" Institute of Numerical Analysis
P.O. Box 68-1, 3400
Cluj-Napoca, Romania
e-mail: cmustata@ictp-acad.math.ubbcluj.ro
2001

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