Extension of starshaped bounded semi-Lipschitz function on a quasi metric linear spaces

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

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C. Mustăţa, Extension of starshaped bounded semi-Lipschitz function on a quasi metric linear spaces. ”135-th Pannonian Applied Mathematical Meeting” 4-7 oct. 2001, 285-291.

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2001-Mustata-BAM-Extension-of-bounded-starhaped-semi-Lipschitz-functions-on-quasi-metric-linear-spac

Extension of bounded starshaped semi-Lipschitz functions on quasi-metric linear spaces

Costică Mustăţa

1 Introduction

Let X X XXX be a nonvoid 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,) satisfying the conditions:
(i) d ( x , y ) = 0 x = y d ( x , y ) = 0 x = y quad d(x,y)=0Longleftrightarrow x=y\quad d(x, y)=0 \Longleftrightarrow x=yd(x,y)=0x=y,
(ii) d ( x , y ) d ( x , z ) + d ( z , y ) d ( x , y ) d ( x , z ) + d ( z , y ) quad d(x,y) <= d(x,z)+d(z,y)\quad 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,zX is called a quasi-metric on X X XXX and the pair ( X , d ) ( X , d ) (X,d)(X, d)(X,d) is called quasi-metric space. The essential difference with respect to a metric on X X XXX is that a quasi-metric does not satisfy the symmetry condition d ( x , y ) = d ( y , x ) d ( x , y ) = d ( y , x ) d(x,y)=d(y,x)d(x, y)= d(y, x)d(x,y)=d(y,x).
If X X XXX is a linear space and d d ddd 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 linear space.
Let θ X θ X theta in X\theta \in XθX be the null element of the linear space X X XXX. A subset Y Y YYY of X X XXX is called starshaped (with respect to θ θ theta\thetaθ ) if it satisfies the condition:
(1) y Y α [ 0 , 1 ] : α y Y . (1) y Y α [ 0 , 1 ] : α y Y . {:(1)AA y in Y quad AA alpha in[0","1]:alpha y in Y.:}\begin{equation*} \forall y \in Y \quad \forall \alpha \in[0,1]: \alpha y \in Y . \tag{1} \end{equation*}(1)yYα[0,1]:αyY.
If Y Y YYY is a starshaped subset of the linear space X X XXX then a function f : Y R f : Y R f:Y rarrRf: Y \rightarrow \mathbb{R}f:YR is called a starshaped function provided:
(2) y Y α [ 0 , 1 ] : f ( α y ) α f ( y ) . (2) y Y α [ 0 , 1 ] : f ( α y ) α f ( y ) . {:(2)AA y in Y quad AA alpha in[0","1]:f(alpha y) <= alpha f(y).:}\begin{equation*} \forall y \in Y \quad \forall \alpha \in[0,1]: f(\alpha y) \leq \alpha f(y) . \tag{2} \end{equation*}(2)yYα[0,1]:f(αy)αf(y).
Obviously that the condition (1) implies θ Y θ Y theta in Y\theta \in YθY and the condition (2) implies f ( θ ) 0 f ( θ ) 0 f(theta) <= 0f(\theta) \leq 0f(θ)0. In what follows we shall consider only starshaped functions on Y Y YYY which vanish at θ θ theta\thetaθ, i.e. f ( θ ) = 0 f ( θ ) = 0 f(theta)=0f(\theta)=0f(θ)=0.
If ( X , d X , d X,dX, dX,d ) is a quasi-metric linear space and Y X Y X Y sub XY \subset XYX is a starshaped set, the quasi-metric d d ddd is called starshaped on Y Y YYY if
(3) x , y Y , α [ 0 , 1 ] : d ( α x , α y ) α d ( x , y ) . (3) x , y Y , α [ 0 , 1 ] : d ( α x , α y ) α d ( x , y ) . {:(3)AA x","y in Y","quad AA alpha in[0","1]:quad d(alpha x","alpha y) <= alpha d(x","y).:}\begin{equation*} \forall x, y \in Y, \quad \forall \alpha \in[0,1]: \quad d(\alpha x, \alpha y) \leq \alpha d(x, y) . \tag{3} \end{equation*}(3)x,yY,α[0,1]:d(αx,αy)αd(x,y).
Let ( X , d ) ( X , d ) (X,d)(X, d)(X,d) be a quasi-metric space and Y X , Y Y X , Y Y sub X,Y!=O/Y \subset X, Y \neq \emptysetYX,Y. A function f : Y R f : Y R f:Y rarrRf: Y \rightarrow \mathbb{R}f:YR is called semi-Lipschitz if it satisfies the condition
(4) K Y 0 : f ( x ) f ( y ) K Y d ( x , y ) (4) K Y 0 : f ( x ) f ( y ) K Y d ( x , y ) {:(4)EEK_(Y) >= 0:f(x)-f(y) <= K_(Y)*d(x","y):}\begin{equation*} \exists K_{Y} \geq 0: f(x)-f(y) \leq K_{Y} \cdot d(x, y) \tag{4} \end{equation*}(4)KY0:f(x)f(y)KYd(x,y)
for all x , y Y x , y Y x,y in Yx, y \in Yx,yY.
A number K Y 0 K Y 0 K_(Y) >= 0K_{Y} \geq 0KY0 for which (4) holds is called a semi-Lipschitz constant for f f fff (on Y Y YYY ).
One sees that
(5) f Y = sup { ( f ( x ) f ( y ) ) 0 d ( x , y ) : x , y Y , d ( x , y ) > 0 } (5) f Y = sup ( f ( x ) f ( y ) ) 0 d ( x , y ) : x , y Y , d ( x , y ) > 0 {:(5)||f||_(Y)=s u p{((f(x)-f(y))vv0)/(d(x,y)):x,y in Y,d(x,y) > 0}:}\begin{equation*} \|f\|_{Y}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: x, y \in Y, d(x, y)>0\right\} \tag{5} \end{equation*}(5)fY=sup{(f(x)f(y))0d(x,y):x,yY,d(x,y)>0}
is the smallest semi-Lipschitz constant for the function f f fff on the set Y Y YYY (see [ 6 ] [ 6 ] [6][6][6] and [4, Th.1]).
Let
(6) S L i p Y := { f : Y R , f is semi-Lipschitz } (6) S L i p Y := { f : Y R , f  is semi-Lipschitz  } {:(6)SLipY:={f:Y rarrR","f" is semi-Lipschitz "}:}\begin{equation*} S L i p Y:=\{f: Y \rightarrow \mathbb{R}, f \text { is semi-Lipschitz }\} \tag{6} \end{equation*}(6)SLipY:={f:YR,f is semi-Lipschitz }
the set of all real-valued semi-Lipschitz functions defined on the quasi-metric space ( Y , d ) , Y X ( Y , d ) , Y X (Y,d),Y sube X(Y, d), Y \subseteq X(Y,d),YX.
If ( X , d X , d X,dX, dX,d ) is a quasi-metric linear space and Y Y YYY is a subset of X X XXX containing θ θ theta\thetaθ then the set
(7) SLip 0 Y := { f f S Lip Y and f ( θ ) = 0 } (7) SLip 0 Y := { f f S Lip Y  and  f ( θ ) = 0 } {:(7)SLip_(0)Y:={f∣f in S Lip Y" and "f(theta)=0}:}\begin{equation*} \operatorname{SLip}_{0} Y:=\{f \mid f \in S \operatorname{Lip} Y \text { and } f(\theta)=0\} \tag{7} \end{equation*}(7)SLip0Y:={ffSLipY and f(θ)=0}
is a semilinear space and the functional Y : S Y : S ||*||_(Y):S\|\cdot\|_{Y}: SY:S Lip 0 Y R + 0 Y R + _(0)Y rarrR_(+)_{0} Y \rightarrow \mathbb{R}_{+}0YR+defined by
(8) f Y : sup { ( f ( x ) f ( y ) ) 0 d ( x , y ) : x , y Y , d ( x , y ) 0 } (8) f Y : sup ( f ( x ) f ( y ) ) 0 d ( x , y ) : x , y Y , d ( x , y ) 0 {:(8)||f||_(Y):s u p{((f(x)-f(y))vv0)/(d(x,y)):x,y in Y,d(x,y)!=0}:}\begin{equation*} \|f\|_{Y}: \sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: x, y \in Y, d(x, y) \neq 0\right\} \tag{8} \end{equation*}(8)fY:sup{(f(x)f(y))0d(x,y):x,yY,d(x,y)0}
is a quasi-norm on S L i p 0 X S L i p 0 X SLip_(0)XS L i p_{0} XSLip0X (see [4] and [6]).
A semilinear space satisfies axioms similar to those defining a linear space, excepting the existence of the opposite element (the inverse with respect to + + ^(+){ }^{+}+) and that the multiplication is defined only for positive scalars. (see [6]).

2 Extensions

The following problem is treated in [4]:
Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, Y X Y X Y sub XY \subset XYX and f S f S f in Sf \in SfS Lip Y Y YYY. One asks to find a function F S Lip X F S Lip X F in S Lip XF \in S \operatorname{Lip} XFSLipX such that
(9) F | Y = f and F X = f Y . (9) F Y = f  and  F X = f Y . {:(9)F|_(Y)=f" and "||F||_(X)=||f||_(Y).:}\begin{equation*} \left.F\right|_{Y}=f \text { and }\|F\|_{X}=\|f\|_{Y} . \tag{9} \end{equation*}(9)F|Y=f and FX=fY.
One shows ([4, Th.2]) that for every f S f S f in Sf \in SfS LipY there exists F S F S F in SF \in SFS LipX satisfying (9). This result is similar to a result of Mc Shane from 1934 (see [2]) asserting that every real valued Lipschitz function defined on a subset of a metric space X X XXX admits an extension to the whole space with the same Lipschitz constant.
In the present paper we study the extension problem for bounded starshaped semi-Lipschitz functions defined on sharshaped subsets of quasi-metric linear spaces:
For a quasi-metric linear space ( X , d ) ( X , d ) (X,d)(X, d)(X,d) with the quasi-metric d d ddd starshaped, a starshaped subset Y Y YYY of X X XXX and a bounded semi-Lipschitz starshaped function f SLip 0 Y f SLip 0 Y f inSLip_(0)Yf \in \operatorname{SLip}_{0} YfSLip0Y find a bounded starshaped function F SLip 0 X F SLip 0 X F inSLip_(0)XF \in \operatorname{SLip}_{0} XFSLip0X such that
(10) F | Y = f , F X = f Y and F = f (10) F Y = f , F X = f Y  and  F = f {:(10)F|_(Y)=f","||F||_(X)=||f||_(Y)" and "||F||_(oo)=||f||_(oo):}\begin{equation*} \left.F\right|_{Y}=f,\|F\|_{X}=\|f\|_{Y} \text { and }\|F\|_{\infty}=\|f\|_{\infty} \tag{10} \end{equation*}(10)F|Y=f,FX=fY and F=f
where ||*||_(oo)\|\cdot\|_{\infty} denotes the sup-norm.
Observe that if Y Y YYY is a subspace of X X XXX and f SLip 0 Y f SLip 0 Y f inSLip_(0)Yf \in \operatorname{SLip}_{0} YfSLip0Y is starshaped then it is possible that f f fff be unbounded on Y Y YYY.
More exactly we have:
Lemma 1 Let ( X , d ) ( X , d ) (X,d)(X, d)(X,d) be a quasi-metric linear space, Y Y YYY a subspace of X X XXX and f f f inf \inf SLip 0 Y 0 Y _(0)Y_{0} Y0Y be starshaped. If there exists x 0 Y x 0 Y x_(0)in Yx_{0} \in Yx0Y such that f ( x 0 ) > 0 f x 0 > 0 f(x_(0)) > 0f\left(x_{0}\right)>0f(x0)>0 then f f fff is unbounded on Y Y YYY.
Proof. For every x Y , x 0 x Y , x 0 x in Y,x!=0x \in Y, x \neq 0xY,x0, the function h : ( 0 , ) R h : ( 0 , ) R h:(0,oo)rarrRh:(0, \infty) \rightarrow \mathbb{R}h:(0,)R defined by h ( t ) = f ( t x ) / t h ( t ) = f ( t x ) / t h(t)=f(tx)//th(t)=f(t x) / th(t)=f(tx)/t is non-increasing. Indeed, if 0 < t 1 < t 2 0 < t 1 < t 2 0 < t_(1) < t_(2)0<t_{1}<t_{2}0<t1<t2 then
h ( t 1 ) = f ( t 1 x ) t 1 = f ( t 2 1 t 1 ( t 2 x ) ) t 1 t 1 t 2 f ( t 2 x ) t 1 = f ( t 2 x ) t 2 = h ( t 2 ) . h t 1 = f t 1 x t 1 = f t 2 1 t 1 t 2 x t 1 t 1 t 2 f t 2 x t 1 = f t 2 x t 2 = h t 2 . h(t_(1))=(f(t_(1)x))/(t_(1))=(f(t_(2)^(-1)*t_(1)(t_(2)x)))/(t_(1)) <= (t_(1))/(t_(2))(f(t_(2)x))/(t_(1))=(f(t_(2)x))/(t_(2))=h(t_(2)).h\left(t_{1}\right)=\frac{f\left(t_{1} x\right)}{t_{1}}=\frac{f\left(t_{2}^{-1} \cdot t_{1}\left(t_{2} x\right)\right)}{t_{1}} \leq \frac{t_{1}}{t_{2}} \frac{f\left(t_{2} x\right)}{t_{1}}=\frac{f\left(t_{2} x\right)}{t_{2}}=h\left(t_{2}\right) .h(t1)=f(t1x)t1=f(t21t1(t2x))t1t1t2f(t2x)t1=f(t2x)t2=h(t2).
In particular, for t > 1 t > 1 t > 1t>1t>1 and x 0 Y x 0 Y x_(0)in Yx_{0} \in Yx0Y with f ( x 0 ) > 0 f x 0 > 0 f(x_(0)) > 0f\left(x_{0}\right)>0f(x0)>0 we have
0 < f ( x 0 ) f ( t x 0 ) t 0 < f x 0 f t x 0 t 0 < f(x_(0)) <= (f(tx_(0)))/(t)0<f\left(x_{0}\right) \leq \frac{f\left(t x_{0}\right)}{t}0<f(x0)f(tx0)t
so that
f ( t x 0 ) t f ( x 0 ) f t x 0 t f x 0 f(tx_(0)) >= tf(x_(0))f\left(t x_{0}\right) \geq t f\left(x_{0}\right)f(tx0)tf(x0)
for every t > 1 t > 1 t > 1t>1t>1, which shows that f f fff is unbounded on Y Y YYY.
The following theorem answers positively the question on the extension of bounded starshaped semi-Lipschitz functions.
Theorem 2 Let ( X , d X , d X,dX, dX,d ) be a quasi-metric linear space and Y Y YYY a starshaped subset of X X XXX. Suppose that the quasi-metric d d ddd is starshaped on Y Y YYY.
Let f S f S f in Sf \in SfS Lip 0 Y 0 Y _(0)Y_{0} Y0Y be bounded and starshaped. In order to exist a bounded starshaped function F S L i p 0 X F S L i p 0 X F in SLip_(0)XF \in S L i p_{0} XFSLip0X such that
F | Y = f , F X = f Y and F = f F Y = f , F X = f Y  and  F = f F|_(Y)=f,quad||F||_(X)=||f||_(Y)quad" and "quad||F||_(oo)=||f||_(oo)\left.F\right|_{Y}=f, \quad\|F\|_{X}=\|f\|_{Y} \quad \text { and } \quad\|F\|_{\infty}=\|f\|_{\infty}F|Y=f,FX=fY and F=f
it is necessary and sufficient that f ( y ) 0 f ( y ) 0 f(y) <= 0f(y) \leq 0f(y)0 for all y Y y Y y in Yy \in YyY.
Proof. Sufficiency suppose that f ( y ) 0 f ( y ) 0 f(y) <= 0f(y) \leq 0f(y)0 for all y Y y Y y in Yy \in YyY. The function
(11) H ( x ) = inf y Y [ f ( y ) + f Y d ( x , y ) ] , x X (11) H ( x ) = inf y Y f ( y ) + f Y d ( x , y ) , x X {:(11)H(x)=i n f_(y in Y)[f(y)+||f||_(Y)d(x,y)]","quad x in X:}\begin{equation*} H(x)=\inf _{y \in Y}\left[f(y)+\|f\|_{Y} d(x, y)\right], \quad x \in X \tag{11} \end{equation*}(11)H(x)=infyY[f(y)+fYd(x,y)],xX
is starshaped on X X XXX and satisfies the conditions
H | Y = f and H X = f Y H Y = f  and  H X = f Y H|_(Y)=f quad" and "quad||H||_(X)=||f||_(Y)\left.H\right|_{Y}=f \quad \text { and } \quad\|H\|_{X}=\|f\|_{Y}H|Y=f and HX=fY
(see [4,Th.2] and [5, Th.8).
Indeed, let z Y z Y z in Yz \in YzY and x X x X x in Xx \in XxX. For any y Y y Y y in Yy \in YyY 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 \\ & \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)+fYd(x,y)=f(z)+fYd(x,y)(f(z)f(y))f(z)+fYd(x,y)fYd(z,y)==f(z)fY(d(z,y)d(x,y))
The inequality d ( z , y ) d ( x , y ) d ( z , x ) d ( z , y ) d ( x , y ) d ( z , x ) d(z,y)-d(x,y) <= d(z,x)d(z, y)-d(x, y) \leq d(z, x)d(z,y)d(x,y)d(z,x) implies
f ( y ) + f Y d ( x , y ) f ( z ) f Y d ( z , x ) f ( y ) + f Y d ( x , y ) f ( z ) f Y d ( z , x ) f(y)+||f||_(Y)d(x,y) >= f(z)-||f||_(Y)*d(z,x)f(y)+\|f\|_{Y} d(x, y) \geq f(z)-\|f\|_{Y} \cdot d(z, x)f(y)+fYd(x,y)f(z)fYd(z,x)
showing that for every x X x X x in Xx \in XxX 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)+fYd(x,y):yY} is bounded from above by f ( z ) f Y d ( z , x ) f ( z ) f Y d ( z , x ) f(z)-||f||_(Y)d(z,x)f(z)-\|f\|_{Y} d(z, x)f(z)fYd(z,x), and the infimum (11) is finite.
We show now that H ( y ) = f ( y ) H ( y ) = f ( y ) H(y)=f(y)H(y)=f(y)H(y)=f(y) for all y Y y Y y in Yy \in YyY.
Let y Y y Y y in Yy \in YyY. Then
H ( y ) f ( y ) + f Y d ( y , y ) = f ( y ) H ( y ) f ( y ) + f Y d ( y , y ) = f ( y ) H(y) <= f(y)+||f||_(Y)d(y,y)=f(y)H(y) \leq f(y)+\|f\|_{Y} d(y, y)=f(y)H(y)f(y)+fYd(y,y)=f(y)
For any v Y v Y v in Yv \in YvY 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)fYd(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)+fYd(y,v)f(y)
and
H ( y ) = inf { f ( v ) + f Y d ( y , v ) : v Y } f ( y ) . H ( y ) = inf f ( v ) + f Y d ( y , v ) : v Y f ( y ) . H(y)=i n f{f(v)+||f||_(Y)d(y,v):v in Y} >= f(y).H(y)=\inf \left\{f(v)+\|f\|_{Y} d(y, v): v \in Y\right\} \geq f(y) .H(y)=inf{f(v)+fYd(y,v):vY}f(y).
It follows H ( y ) = f ( y ) H ( y ) = f ( y ) H(y)=f(y)H(y)=f(y)H(y)=f(y).
We prove that H Y = f Y H Y = f Y ||H||_(Y)=||f||_(Y)\|H\|_{Y}=\|f\|_{Y}HY=fY.
Since H | Y = f H Y = f H|_(Y)=f\left.H\right|_{Y}=fH|Y=f, the definitions of H Y H Y ||H||_(Y)\|H\|_{Y}HY and f Y f Y ||f||_(Y)\|f\|_{Y}fY yield H Y f Y H Y f Y ||H||_(Y) >= ||f||_(Y)\|H\|_{Y} \geq\|f\|_{Y}HYfY.
Let x 1 , x 2 X x 1 , x 2 X x_(1),x_(2)in Xx_{1}, x_{2} \in Xx1,x2X and ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0. Choosing y Y y Y y in Yy \in YyY such that
H ( x 1 ) f ( y ) + f Y d ( x 1 , y ) ε H x 1 f ( y ) + f Y d x 1 , y ε H(x_(1)) >= f(y)+||f||_(Y)d(x_(1),y)-epsiH\left(x_{1}\right) \geq f(y)+\|f\|_{Y} d\left(x_{1}, y\right)-\varepsilonH(x1)f(y)+fYd(x1,y)ε
we obtain
H ( x 2 ) H ( 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 . H x 2 H 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 . {:[H(x_(2))-H(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} H\left(x_{2}\right)-H\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}H(x2)H(x1)f(y)+fYd(x2,y)(f(y)+fYd(x1,y)ε)=fY[d(x2,y)d(x1,y)]+εfYd(x2,x1)+ε.
Since ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 is arbitrary it follows
H ( x 2 ) H ( x 1 ) f Y d ( x 2 , x 1 ) H x 2 H x 1 f Y d x 2 , x 1 H(x_(2))-H(x_(1)) <= ||f||_(Y)*d(x_(2),x_(1))H\left(x_{2}\right)-H\left(x_{1}\right) \leq\|f\|_{Y} \cdot d\left(x_{2}, x_{1}\right)H(x2)H(x1)fYd(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,x2X and H Y f Y H Y f Y ||H||_(Y) <= ||f||_(Y)\|H\|_{Y} \leq\|f\|_{Y}HYfY. Then H X = f Y H X = f Y ||H||_(X)=||f||_(Y)\|H\|_{X}=\|f\|_{Y}HX=fY.
We shall show that H H HHH is also starshaped on X X XXX. To this end let x X , z Y x X , z Y x in X,z in Yx \in X, z \in YxX,zY and α [ 0 , 1 ] α [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α[0,1]. We have
H ( α x ) = f ( α z ) + f Y d ( α x , α z ) α f ( z ) + α f Y d ( x , z ) = = α [ f ( z ) + f Y d ( x , z ) ] . H ( α x ) = f ( α z ) + f Y d ( α x , α z ) α f ( z ) + α f Y d ( x , z ) = = α f ( z ) + f Y d ( x , z ) . {:[H(alpha x)=f(alpha z)+||f||_(Y)d(alpha x","alpha z) <= ],[ <= alpha f(z)+alpha||f||_(Y)d(x","z)=],[=alpha[f(z)+||f||_(Y)*d(x,z)].]:}\begin{aligned} H(\alpha x) & =f(\alpha z)+\|f\|_{Y} d(\alpha x, \alpha z) \leq \\ & \leq \alpha f(z)+\alpha\|f\|_{Y} d(x, z)= \\ & =\alpha\left[f(z)+\|f\|_{Y} \cdot d(x, z)\right] . \end{aligned}H(αx)=f(αz)+fYd(αx,αz)αf(z)+αfYd(x,z)==α[f(z)+fYd(x,z)].
Taking the infimum with respect to z Y z Y z in Yz \in YzY we get
H ( α x ) α H ( x ) H ( α x ) α H ( x ) H(alpha x) <= alpha H(x)H(\alpha x) \leq \alpha H(x)H(αx)αH(x)
for all x X x X x in Xx \in XxX and all α [ 0 , 1 ] α [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α[0,1], showing that the function H H HHH defined by (11) is a starshaped extension of f f fff.
Consider the function
(12) F ( x ) = { H ( x ) if H ( x ) 0 0 if H ( x ) > 0 . (12) F ( x ) = H ( x )  if  H ( x ) 0 0  if  H ( x ) > 0 . {:(12)F(x)={[H(x)," if ",H(x) <= 0],[0," if ",H(x) > 0].:}:}F(x)=\left\{\begin{array}{lll} H(x) & \text { if } & H(x) \leq 0 \tag{12}\\ 0 & \text { if } & H(x)>0 \end{array} .\right.(12)F(x)={H(x) if H(x)00 if H(x)>0.
Since F | Y = H | Y = f , H X = f Y F Y = H Y = f , H X = f Y F|_(Y)=H|_(Y)=f,||H||_(X)=||f||_(Y)\left.F\right|_{Y}=\left.H\right|_{Y}=f,\|H\|_{X}=\|f\|_{Y}F|Y=H|Y=f,HX=fY and H X F X H X F X ||H||_(X) >= ||F||_(X)\|H\|_{X} \geq\|F\|_{X}HXFX, it follows that F X = f Y F X = f Y ||F||_(X)=||f||_(Y)\|F\|_{X}=\|f\|_{Y}FX=fY. Therefore F F FFF is an extension of f f fff with the same semi-Lipschitz constant.
Let x X x X x in Xx \in XxX. If H ( x ) 0 H ( x ) 0 H(x) <= 0H(x) \leq 0H(x)0 then
H ( α x ) α H ( x ) 0 H ( α x ) α H ( x ) 0 H(alpha x) <= alpha H(x) <= 0H(\alpha x) \leq \alpha H(x) \leq 0H(αx)αH(x)0
for any α [ 0 , 1 ] α [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α[0,1] so that
F ( α x ) = H ( α x ) α H ( x ) = α F ( x ) . F ( α x ) = H ( α x ) α H ( x ) = α F ( x ) . F(alpha x)=H(alpha x) <= alpha H(x)=alpha F(x).F(\alpha x)=H(\alpha x) \leq \alpha H(x)=\alpha F(x) .F(αx)=H(αx)αH(x)=αF(x).
If H ( x ) > 0 H ( x ) > 0 H(x) > 0H(x)>0H(x)>0 then F ( x ) = 0 F ( x ) = 0 F(x)=0F(x)=0F(x)=0. Let α ( 0 , 1 ) α ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α(0,1). If H ( α x ) > 0 H ( α x ) > 0 H(alpha x) > 0H(\alpha x)>0H(αx)>0 then
F ( α x ) = 0 = α F ( x ) . F ( α x ) = 0 = α F ( x ) . F(alpha x)=0=alpha F(x).F(\alpha x)=0=\alpha F(x) .F(αx)=0=αF(x).
If H ( α x ) 0 H ( α x ) 0 H(alpha x) <= 0H(\alpha x) \leq 0H(αx)0 then F ( α x ) = H ( α x ) F ( α x ) = H ( α x ) F(alpha x)=H(alpha x)F(\alpha x)=H(\alpha x)F(αx)=H(αx) and
F ( α x ) = H ( α x ) 0 = α F ( x ) . F ( α x ) = H ( α x ) 0 = α F ( x ) . F(alpha x)=H(alpha x) <= 0=alpha F(x).F(\alpha x)=H(\alpha x) \leq 0=\alpha F(x) .F(αx)=H(αx)0=αF(x).
It follows that the function F F FFF defined by (12) is an extension of f f fff which is starshaped and has the same semi-Lipschitz constant as f f fff.
Since
sup { | F ( x ) | : x X } sup { | F ( y ) | : y Y } = f sup { | F ( x ) | : x X } sup { | F ( y ) | : y Y } = f s u p{|F(x)|:x in X} >= s u p{|F(y)|:y in Y}=||f||_(oo)\sup \{|F(x)|: x \in X\} \geq \sup \{|F(y)|: y \in Y\}=\|f\|_{\infty}sup{|F(x)|:xX}sup{|F(y)|:yY}=f
it follows F f F f ||F||_(oo) >= ||f||_(oo)\|F\|_{\infty} \geq\|f\|_{\infty}Ff.
Let x X x X x in Xx \in XxX. For any y Y y Y y in Yy \in YyY we have
f ( y ) f ( y ) + f Y d ( x , y ) f ( y ) f ( y ) + f Y d ( x , y ) f(y) <= f(y)+||f||_(Y)*d(x,y)f(y) \leq f(y)+\|f\|_{Y} \cdot d(x, y)f(y)f(y)+fYd(x,y)
which implies
inf y Y f ( y ) inf y Y [ f ( y ) + f Y d ( x , y ) ] = H ( x ) . inf y Y f ( y ) inf y Y f ( y ) + f Y d ( x , y ) = H ( x ) . i n f_(y inY^('))f(y) <= i n f_(y in Y)[f(y)+||f||_(Y)d(x,y)]=H(x).\inf _{y \in Y^{\prime}} f(y) \leq \inf _{y \in Y}\left[f(y)+\|f\|_{Y} d(x, y)\right]=H(x) .infyYf(y)infyY[f(y)+fYd(x,y)]=H(x).
If H ( x ) 0 H ( x ) 0 H(x) <= 0H(x) \leq 0H(x)0 then
inf y Y f ( y ) H ( x ) = F ( x ) . inf y Y f ( y ) H ( x ) = F ( x ) . i n f_(y in Y)f(y) <= H(x)=F(x).\inf _{y \in Y} f(y) \leq H(x)=F(x) .infyYf(y)H(x)=F(x).
Since f ( y ) 0 f ( y ) 0 f(y) <= 0f(y) \leq 0f(y)0 for all y Y y Y y in Yy \in YyY we have f 1 2 = inf f ( Y ) f 1 2 = inf f ( Y ) ||f||_(oo)^((1)/(2))=-i n f f(Y)\|f\|_{\infty}^{\frac{1}{2}}=-\inf f(Y)f12=inff(Y) so that
f F ( x ) f F ( x ) . f F ( x ) f F ( x ) . -||f||_(oo) <= F(x)Longleftrightarrow||f||_(oo) >= -F(x).-\|f\|_{\infty} \leq F(x) \Longleftrightarrow\|f\|_{\infty} \geq-F(x) .fF(x)fF(x).
If H ( x ) > 0 H ( x ) > 0 H(x) > 0H(x)>0H(x)>0 then
f 0 = F ( x ) . f 0 = F ( x ) . ||f||_(oo) >= 0=F(x).\|f\|_{\infty} \geq 0=F(x) .f0=F(x).
Using the fact that F ( x ) 0 F ( x ) 0 F(x) <= 0F(x) \leq 0F(x)0, for all x X x X x in Xx \in XxX. and the above inequalities we obtain
f sup x X ( F ( x ) ) = F . f sup x X ( F ( x ) ) = F . ||f||_(oo) >= s u p_(x in X)(-F(x))=||F||_(oo).\|f\|_{\infty} \geq \sup _{x \in X}(-F(x))=\|F\|_{\infty} .fsupxX(F(x))=F.
It follows that F = f F = f ||F||_(oo)=||f||_(oo)\|F\|_{\infty}=\|f\|_{\infty}F=f.
Necessity. Suppose there exists y Y y Y y in Yy \in YyY such that f ( y ) > 0 f ( y ) > 0 f(y) > 0f(y)>0f(y)>0. By Lemma 1 f f fff has no bounded starshaped extensions to X X XXX.

References

[1] Cobzaş, S., Mustăta, C., Norm preserving extension of convex Lipschitz functions, J.A.T. 24(1978), 555-564.
[2] McShane, J.A., Extension of range of functions, Bull. Amer. Math. Soc. 40(1934), 837-842.
[3] Mustăta, C., The extension of starshaped bounded Lipschitz functions, Ann. Numer. Theor. Approx., 9(1980) 1, 93-99.
[4] Mustăta, C., Extensions of semi-Lipschitz functions on quasi-metric space (to appear).
[5] Mustăta, C., Extension of convex semi-Lipschitz functions on quasimetric linear space (to appear)
[6] Romaguera, S., Sanchis, M., Semi-Lipschitz functions in quasi-metric spaces, J.A.T 103(2000), 292-301.
[7] Weaver, N., Lipschitz Algebras, World Scientific, Singapore 1999.
[8] Wels, J.H., Williams, L.R., Embeddings and Extension in Analysis, Springer-Verlag, Berlin, 1975.
2000

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