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.
Paper (preprint) in HTML form
Extensions of convex semi-Lipschitz functions on quasi-metric linear spaces
Abstract
In this paper one shows that a convex semi-Lipschitz functions defined on a convex subset of a quasi-metric linear space XX admits an extension to the whole space XX, preserving both the convexity and the semi-Lipschitz constant. A similar result is proved for starshaped functions.
1 Introduction
Let XX be a set. A function d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty) is called a quasi-metric if
(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),
for all x.y.z in Xx . y . z \in X. If dd is a quasi-metric on XX then the pair (X,d)(X, d) is called a quasi-metric space. If XX is further a linear space and dd is a quasi-metric on XX then the pair ( X,dX, d ) is called a quasi-metric linear space.
(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),
for all x.y.z in Xx . y . z \in X. If dd is a quasi-metric on XX then the pair (X,d)(X, d) is called a quasi-metric space. If XX is further a linear space and dd is a quasi-metric on XX then the pair ( X,dX, d ) is called a quasi-metric linear space.
The function d^(-1):X xx X rarr[0,oo)d^{-1}: X \times X \rightarrow[0, \infty) defined by equality
{:(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*}
is called the conjugate of the quasi-metric d[1]d[1].
Definition 1.1 Let ( X,dX, d ) be a quasi-metric linear space and YY a convex subset of XX (i.e. (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 ). The quasi-metric dd is called convex on YY if it satisfies the inequality
(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), for all x_(1),x_(2),y_(1),y_(2)in Yx_{1}, x_{2}, y_{1}, y_{2} \in Y and all alpha in[0,1]\alpha \in[0,1].
(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), for all x_(1),x_(2),y_(1),y_(2)in Yx_{1}, x_{2}, y_{1}, y_{2} \in Y and all alpha in[0,1]\alpha \in[0,1].
Definition 1.2 [4]. Let ( X,dX, d ) be a quasi-metric linear space. A function f:X rarrRf: X \rightarrow \mathbb{R} is called semi-Lipschitz if there exists K >= 0K \geq 0 such that
{:(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*}
for all x,y in Xx, y \in X.
A number K >= 0K \geq 0 for which (3) holds is called a semi-Lipschitz constant for ff.
A number K >= 0K \geq 0 for which (3) holds is called a semi-Lipschitz constant for ff.
For Y sub XY \subset X let
(4) 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\}.
(4) 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\}.
The ||f||_(Y)\|f\|_{Y} is the smallest semi-Lipschitz constant of the function ff on YY [3, Th.1].
For Y sub XY \subset X let
{:(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*}
be the set of all real-valued semi-Lipschitz functions defined on the quasi-metric space (Y,d)(Y, d).
Definition 1.3 A function F in SF \in S Lip XX is called an extension of a function f in Sf \in S Lip YY if
(i) F|_(Y)=f\left.F\right|_{Y}=f,
(ii) ||F||_(X)=||f||_(Y)\|F\|_{X}=\|f\|_{Y}.
(i) F|_(Y)=f\left.F\right|_{Y}=f,
(ii) ||F||_(X)=||f||_(Y)\|F\|_{X}=\|f\|_{Y}.
For f in S Lip Yf \in S \operatorname{Lip} Y one denotes by
{:(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*}
the set of all extensions of the function ff.
By Theorem 2 in [4] it follows that
By Theorem 2 in [4] it follows that
{:(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*}
The following problem arises naturally: which other properties of the function ff (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,dX, d ) be a quasi-metric linear space and YY a convex subset of XX. Suppose that the quasi-metric dd is convex on YY, in the sense of Definition 1.
a) If f in Sf \in S Lip YY is convex on YY then there exists a convex
a) If f in Sf \in S Lip YY is convex on YY then there exists a convex
F inE_(Y)(f).F \in E_{Y}(f) .
b) If f in Sf \in S Lip YY is concave (i.e. -f-f is convex) on YY then there exists a concave
G inE_(Y)(f).G \in E_{Y}(f) .
Proof. a) Let f in S Lip Yf \in S \operatorname{Lip} Y be convex on the convex set YY. Consider the function F:X rarrRF: X \rightarrow \mathbb{R} defined by
{:(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*}
Then FF is well defined and F inE_(Y)(f)F \in E_{Y}(f) (Theorem 2 in [4]). Indeed, let z in Yz \in Y and x in Xx \in X. For any y in Yy \in 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)).]:}\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}
The inequality 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) implies
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)
showing that for every x in Xx \in X the set {f(y)+||f||_(Y)d(x,y):y in Y}\left\{f(y)+\|f\|_{Y} d(x, y): y \in Y\right\} is bounded from above by f(z)-||f||_(Y)d^(-1)(x,z)f(z)-\|f\|_{Y} d^{-1}(x, z), and the infimum (8) is finite.
We show now that F(y)=f(y)F(y)=f(y) for all y in Yy \in Y.
Let y in Yy \in Y. Then
Let y in Yy \in Y. Then
F(y) <= f(y)+||f||_(Y)d(y,y)=f(y).F(y) \leq f(y)+\|f\|_{Y} d(y, y)=f(y) .
For any v in Yv \in Y we have
f(y)-f(v) <= ||f||_(Y)*d(y,v)f(y)-f(v) \leq\|f\|_{Y} \cdot d(y, v)
so that
f(v)+||f||_(Y)*d(y,v) >= f(y)f(v)+\|f\|_{Y} \cdot d(y, v) \geq f(y)
and
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)
It follows F(y)=f(y)F(y)=f(y).
We prove that ||F||_(Y)=||f||_(Y)\|F\|_{Y}=\|f\|_{Y}.
Since F|_(Y)=f\left.F\right|_{Y}=f, the definitions of ||F||_(Y)\|F\|_{Y} and ||f||_(Y)\|f\|_{Y} yield ||F||_(Y) >= ||f||_(Y)\|F\|_{Y} \geq \|f\|_{Y}.
We prove that ||F||_(Y)=||f||_(Y)\|F\|_{Y}=\|f\|_{Y}.
Since F|_(Y)=f\left.F\right|_{Y}=f, the definitions of ||F||_(Y)\|F\|_{Y} and ||f||_(Y)\|f\|_{Y} yield ||F||_(Y) >= ||f||_(Y)\|F\|_{Y} \geq \|f\|_{Y}.
Let x_(1),x_(2)in Xx_{1}, x_{2} \in X and epsi > 0\varepsilon>0. Choosing y in Yy \in Y such that
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)-\varepsilon
we obtain
{:[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}
Since epsi > 0\varepsilon>0 is arbitrary it follows
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)
for any x_(1),x_(2)in Xx_{1}, x_{2} \in X and ||F||_(Y) <= ||f||_(Y)\|F\|_{Y} \leq\|f\|_{Y}. Then ||F||_(X)=||f||_(Y)\|F\|_{X}=\|f\|_{Y}.
Since YY is convex and the quasi-metric dd is convex (in the sense
of Definition 1) we have
Since YY is convex and the quasi-metric dd is convex (in the sense
of Definition 1) we have
{:[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}
for all x_(1),x_(2)in X,y_(1),y_(2)in Yx_{1}, x_{2} \in X, y_{1}, y_{2} \in Y and alpha in[0,1]\alpha \in[0,1].
Taking the infimum with respect to y_(1),y_(2)in Yy_{1}, y_{2} \in Y we obtain
Taking the infimum with respect to y_(1),y_(2)in Yy_{1}, y_{2} \in Y we obtain
{:(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*}
for all x_(1),x_(2)in Xx_{1}, x_{2} \in X and all alpha in[0,1]\alpha \in[0,1], showing that the function FF in E_(Y)(f)E_{Y}(f), defined by (8), is convex.
b) If f in Sf \in S Lip YY is concave on YY, let G:X rarrRG: X \rightarrow \mathbb{R} be defined by
b) If f in Sf \in S Lip YY is concave on YY, let G:X rarrRG: X \rightarrow \mathbb{R} be defined by
{:(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*}
Then GG is well defined and G inE_(Y)(f)G \in E_{Y}(f) [4, Theorem 2].
For any x_(1),x_(2)in X,y_(1),y_(2)in Yx_{1}, x_{2} \in X, y_{1}, y_{2} \in Y and alpha in[0,1]\alpha \in[0,1] we have
For any x_(1),x_(2)in X,y_(1),y_(2)in Yx_{1}, x_{2} \in X, y_{1}, y_{2} \in Y and alpha in[0,1]\alpha \in[0,1] we have
{:[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}
Taking the supremum with respect to y_(1),y_(2)in Yy_{1}, y_{2} \in Y we get
{:(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*}
for all x_(1),x_(2)in Xx_{1}, x_{2} \in X and alpha in[0,1]\alpha \in[0,1], showing that the function GG from E_(Y)(f)E_{Y}(f), defined by (10), is concave.
Definition 1.4 Let XX be a real linear space and theta in Z sub X\theta \in Z \subset X where theta\theta denotes the null element of XX. The set ZZ is called starshaped if
{:(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*}
Obviously that any convex subset of XX which contains theta\theta is starshaped, and the converse is not true in general.
A function f:Z rarrRf: Z \rightarrow \mathbb{R}, where ZZ is a starshaped subset of a linear space XX, is called starshaped if
{:(12a)f(alpha z) <= alpha f(z).:}\begin{equation*}
f(\alpha z) \leq \alpha f(z) . \tag{12a}
\end{equation*}
for all z in Zz \in Z and alpha in[0,1]\alpha \in[0,1]. A convex function f:Y rarrRf: Y \rightarrow \mathbb{R}, defined on a convex subset YY of XX containing theta\theta, and such that f(theta) <= 0f(\theta) \leq 0 is starshaped but there are starshaped functions on YY which are not convex.
Definition 1.5 Let ( X,dX, d ) be a quasi-metric linear space. The quasi-metric dd is called starshaped on XX if the inequality
{:(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*}
holds for all x,y in Xx, y \in X and alpha in[0,1]\alpha \in[0,1].
Remark 1.1 If (X,d)(X, d) is a quasi-metric linear space with convex quasi-metric, then dd is starshaped because the inequality
Remark 1.1 If (X,d)(X, d) is a quasi-metric linear space with convex quasi-metric, then dd is starshaped because the inequality
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)
yields
d(alpha x,alpha y) <= alpha d(x,y)d(\alpha x, \alpha y) \leq \alpha d(x, y)
for all x,y in Xx, y \in X and alpha in[0,1]\alpha \in[0,1].
Now we shall prove the extension result for starshaped semiLipschitz functions.
Now we shall prove the extension result for starshaped semiLipschitz functions.
Theorem 1.2 Let ( X,dX, d ) be a quasi-metric linear space with starshaped quasi-metric dd and let ZZ be a starshaped subset of XX.
Then every starshaped function varphi in S\varphi \in S Lip ZZ admits at least one starshaped extension Phi inE_(Z)(varphi)\Phi \in E_{Z}(\varphi).
Proof. Let varphi in S\varphi \in S Lip ZZ be starshaped on the starshaped set Z sub XZ \subset X. The function
{:(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*}
belongs to E_(Z)(varphi)E_{Z}(\varphi) (Th 2 in [4]).
We shall show that Phi\Phi is also starshaped on XX. To this end let x in X,z in Zx \in X, z \in Z and alpha in[0,1]\alpha \in[0,1]. We have
We shall show that Phi\Phi is also starshaped on XX. To this end let x in X,z in Zx \in X, z \in Z and alpha in[0,1]\alpha \in[0,1]. We have
{:[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}
Taking the infimum with respect to z in Zz \in Z we get
Phi(alpha x) <= alpha Phi(x)\Phi(\alpha x) \leq \alpha \Phi(x)
for all x in Xx \in X and all alpha in[0,1]\alpha \in[0,1], showing that the function Phi\Phi defined by (14) is a starshaped extension of varphi\varphi.
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
[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
e-mail: cmustata@ictp-acad.math.ubbcluj.ro