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C. Mustăţa, A characterization of semichebyshevian sets in a metric space, Anal. Numér. Théor. Approx. 7 (1978) 2, 169-170.
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Mathematica-Revue d’Analyse Numer. Theor.Approx.
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Romanian Academy
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MR # 82j: 41034
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[1] Mustăţa, Costică, On the best approximation in metric spaces. Rev. Anal. Numér. Théor. Approx. 4 (1975), no. 1, 45–50, MR0531660.
[2] Singer, Ivan, Cea mai bună aproximare în spaţii vectoriale normate prin elemente din subspaţii vectoriale. (Romanian) [Best approximation in normed vector spaces by elements of vector subspaces] Editura Academiei Republicii Socialiste România, Bucharest 1967 386 pp., MR0235368.
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A CHARACTERIZATION OF SEMI-CHEBYSHEVIAN SETS IN A METRIC SPACE
In [1] is given the theorems of characterization of Chebyshevian sets in a metric space. The present note is a completation of the paper [1].
Let XX be a metric space with the metric dd, let YY be a nonvoid subset of XX such that x_(0)in Yx_{0} \in Y, where x_(0)x_{0} is an fixed element in XX. Let Lip _(0)X{ }_{0} X be the space of all real Lipschitz functions, defined on XX, endowed with the Lipschitz norm ||||_(X):}\left\|\|_{X}\right. [1].
The set YY is called semi-Chebyshevian if for every x⋐X\\Yx \Subset X \backslash Y there exists at most an element y_(0)in Yy_{0} \in Y such that
{:(1)d(x,y_(0))=i n f{d(x","y):y in Y}=d(x","Y).:}\begin{equation*}
d\left(x, y_{0}\right)=\inf \{d(x, y): y \in Y\}=d(x, Y) . \tag{1}
\end{equation*}
An element y_(0)in Yy_{0} \in Y for that (1) holds is called an element of best approximation of xx, by elements of YY.
THEOREM. If YY is a nonvoid subset of the metric space XX such that x_(0)in Yx_{0} \in Y, then the following two asertions are equivalent:
1^(@)Y1^{\circ} Y is semi-Chebyshevian;
2^(@)2^{\circ} There does not exist f inLip_(0)X,x_(1)in Xf \in \operatorname{Lip}_{0} X, x_{1} \in X and y_(1),y_(2)in Y,y_(1)!=y_(2)y_{1}, y_{2} \in Y, y_{1} \neq y_{2} such that
a) ||f||_(X)=1\|f\|_{X}=1,
b) f|_(Y)=0\left.f\right|_{Y}=0,
c) f(x_(1))=d(x_(1),y_(1))=d(x_(1),y_(2))f\left(x_{1}\right)=d\left(x_{1}, y_{1}\right)=d\left(x_{1}, y_{2}\right).
1^(@)Y1^{\circ} Y is semi-Chebyshevian;
2^(@)2^{\circ} There does not exist f inLip_(0)X,x_(1)in Xf \in \operatorname{Lip}_{0} X, x_{1} \in X and y_(1),y_(2)in Y,y_(1)!=y_(2)y_{1}, y_{2} \in Y, y_{1} \neq y_{2} such that
a) ||f||_(X)=1\|f\|_{X}=1,
b) f|_(Y)=0\left.f\right|_{Y}=0,
c) f(x_(1))=d(x_(1),y_(1))=d(x_(1),y_(2))f\left(x_{1}\right)=d\left(x_{1}, y_{1}\right)=d\left(x_{1}, y_{2}\right).
Proof. Let us suppose that there exists f in Lip_(0)Xf \in \operatorname{Lip}{ }_{0} X, there exist x_(1)in Xx_{1} \in X and y_(1),y_(2)in Y,y_(1)!=y_(2)y_{1}, y_{2} \in Y, y_{1} \neq y_{2} such that the conditions {:a),b),c)\left.\left.\left.a\right), b\right), c\right) hold. Then
{:[f(x_(1))=d(x_(1),y_(1))=f(x_(1))-f(y_(1))=|f(x_(1))-f(y_(1))|","],[f(x_(1))=d(x_(1),y_(2))=f(x_(1))-f(y_(2))=|f(x_(1))-f(y_(2))|","]:}\begin{aligned}
& f\left(x_{1}\right)=d\left(x_{1}, y_{1}\right)=f\left(x_{1}\right)-f\left(y_{1}\right)=\left|f\left(x_{1}\right)-f\left(y_{1}\right)\right|, \\
& f\left(x_{1}\right)=d\left(x_{1}, y_{2}\right)=f\left(x_{1}\right)-f\left(y_{2}\right)=\left|f\left(x_{1}\right)-f\left(y_{2}\right)\right|,
\end{aligned}
and by Theorem 4 in [1] it follows that the elements y_(1),y_(2)y_{1}, y_{2} are two distinct elements of the best approximation for x_(1)x_{1}. Consequently, YY is not semi-Chebyshevian.
If YY is not semi-Chebyshevian, then there exists an element x_(1)in Xx_{1} \in X and the elements y_(1),y_(2)in Y,y_(1)!=y_(2)y_{1}, y_{2} \in Y, y_{1} \neq y_{2} such that
d(x_(1),y_(1))=d(x_(1),y_(2))=d(x_(1),Y).d\left(x_{1}, y_{1}\right)=d\left(x_{1}, y_{2}\right)=d\left(x_{1}, Y\right) .
Then, the function f(x)=d(x,Y),x in Xf(x)=d(x, Y), x \in X is in Lip _(0)X{ }_{0} X and verifies the condition a),b),ca), b), c ). It follows that the asertion 2^(@)2^{\circ} is not fulfilled. The theorem is proved.
REFERENCES
[1] Mustă tt a, C., On the Best Approximation in Metric Spaces, Mathematica - Revue d'Aualyse Numérique et de la Théorie de l'Approximation, L'Analyse Numérique et la Théorie de 1'Approximation, 4, 1, 45-50, (1975)
[2] Singer, I., Cea mai bunà aproximare în spafii vectoriale normate prin elemente din subspatii vectoriale, Anexa II, Ed. Acad. R. S. România, Bucureşti, 1967.
[2] Singer, I., Cea mai bunà aproximare în spafii vectoriale normate prin elemente din subspatii vectoriale, Anexa II, Ed. Acad. R. S. România, Bucureşti, 1967.
Received, 9. II. 1978.