A characterization of semi-chebyshevian sets in a metric space

Costica Mustata
(original), Approximation Theory, best approximation, paper

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Costică Mustăţa
Tiberiu Popoviciu Institute of Numerical Analysis

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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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https://ictp.acad.ro/jnaat/journal/article/view/1978-vol7-no2-art6/Mustata

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Mathematica-Revue d’Analyse Numer. Theor.Approx.

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Romanian Academy

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https://ictp.acad.ro/jnaat/journal/article/view/1978-vol7-no2-art6/

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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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1978-Mustata-A characterization of semichebyshevian sets in a metric space-Jnaat

A CHARACTERIZATION OF SEMI-CHEBYSHEVIAN SETS IN A METRIC SPACE

byCOSTICA MUSTĂTA(Cluj-Napoca)

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 X X XXX be a metric space with the metric d d ddd, let Y Y YYY be a nonvoid subset of X X XXX such that x 0 ∈ Y x 0 ∈ Y x_(0)in Yx_{0} \in Yx0∈Y, where x 0 x 0 x_(0)x_{0}x0 is an fixed element in X X XXX. Let Lip 0 X 0 X _(0)X{ }_{0} X0X be the space of all real Lipschitz functions, defined on X X XXX, endowed with the Lipschitz norm ‖ ‖ X ‖ X ||||_(X):}\left\|\|_{X}\right.‖‖X [1].
The set Y Y YYY is called semi-Chebyshevian if for every x ⋐ X ∖ Y x ⋐ X ∖ Y x⋐X\\Yx \Subset X \backslash Yx⋐X∖Y there exists at most an element y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y such that
(1) d ( x , y 0 ) = inf { d ( x , y ) : y ∈ Y } = d ( x , Y ) . (1) d x , y 0 = inf { d ( x , y ) : y ∈ Y } = d ( x , Y ) . {:(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*}(1)d(x,y0)=inf{d(x,y):y∈Y}=d(x,Y).
An element y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y for that (1) holds is called an element of best approximation of x x xxx, by elements of Y Y YYY.
THEOREM. If Y Y YYY is a nonvoid subset of the metric space X X XXX such that x 0 ∈ Y x 0 ∈ Y x_(0)in Yx_{0} \in Yx0∈Y, then the following two asertions are equivalent:
1 ∘ Y 1 ∘ Y 1^(@)Y1^{\circ} Y1∘Y is semi-Chebyshevian;
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ There does not exist f ∈ Lip 0 X , x 1 ∈ X f ∈ Lip 0 X , x 1 ∈ X f inLip_(0)X,x_(1)in Xf \in \operatorname{Lip}_{0} X, x_{1} \in Xf∈Lip0X,x1∈X and y 1 , y 2 ∈ Y , y 1 ≠ y 2 y 1 , y 2 ∈ Y , y 1 ≠ y 2 y_(1),y_(2)in Y,y_(1)!=y_(2)y_{1}, y_{2} \in Y, y_{1} \neq y_{2}y1,y2∈Y,y1≠y2 such that
a) ‖ f ‖ X = 1 ‖ f ‖ X = 1 ||f||_(X)=1\|f\|_{X}=1‖f‖X=1,
b) f | Y = 0 f Y = 0 f|_(Y)=0\left.f\right|_{Y}=0f|Y=0,
c) f ( x 1 ) = d ( x 1 , y 1 ) = d ( x 1 , y 2 ) f x 1 = d x 1 , y 1 = d x 1 , y 2 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)f(x1)=d(x1,y1)=d(x1,y2).
Proof. Let us suppose that there exists f ∈ Lip 0 X f ∈ Lip 0 X f in Lip_(0)Xf \in \operatorname{Lip}{ }_{0} Xf∈Lip0X, there exist x 1 ∈ X x 1 ∈ X x_(1)in Xx_{1} \in Xx1∈X and y 1 , y 2 ∈ Y , y 1 ≠ y 2 y 1 , y 2 ∈ Y , y 1 ≠ y 2 y_(1),y_(2)in Y,y_(1)!=y_(2)y_{1}, y_{2} \in Y, y_{1} \neq y_{2}y1,y2∈Y,y1≠y2 such that the conditions a ) , b ) , c ) a , b , c {:a),b),c)\left.\left.\left.a\right), b\right), c\right)a),b),c) 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 ) | , 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 , {:[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}f(x1)=d(x1,y1)=f(x1)−f(y1)=|f(x1)−f(y1)|,f(x1)=d(x1,y2)=f(x1)−f(y2)=|f(x1)−f(y2)|,
and by Theorem 4 in [1] it follows that the elements y 1 , y 2 y 1 , y 2 y_(1),y_(2)y_{1}, y_{2}y1,y2 are two distinct elements of the best approximation for x 1 x 1 x_(1)x_{1}x1. Consequently, Y Y YYY is not semi-Chebyshevian.
If Y Y YYY is not semi-Chebyshevian, then there exists an element x 1 ∈ X x 1 ∈ X x_(1)in Xx_{1} \in Xx1∈X and the elements y 1 , y 2 ∈ Y , y 1 ≠ y 2 y 1 , y 2 ∈ Y , y 1 ≠ y 2 y_(1),y_(2)in Y,y_(1)!=y_(2)y_{1}, y_{2} \in Y, y_{1} \neq y_{2}y1,y2∈Y,y1≠y2 such that
d ( x 1 , y 1 ) = d ( x 1 , y 2 ) = d ( x 1 , Y ) . d x 1 , y 1 = d x 1 , y 2 = d x 1 , Y . 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) .d(x1,y1)=d(x1,y2)=d(x1,Y).
Then, the function f ( x ) = d ( x , Y ) , x ∈ X f ( x ) = d ( x , Y ) , x ∈ X f(x)=d(x,Y),x in Xf(x)=d(x, Y), x \in Xf(x)=d(x,Y),x∈X is in Lip 0 X 0 X _(0)X{ }_{0} X0X and verifies the condition a ) , b ) , c a ) , b ) , c a),b),ca), b), ca),b),c ). It follows that the asertion 2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ is not fulfilled. The theorem is proved.

REFERENCES

[1] Mustă t t ttt 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.
Received, 9. II. 1978.

Universilatea ,Babeṣ-Bolyai''Cluj-NapocaInstitutul de Matematică

1978

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