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Difference between revisions of "Chebyshev centre"

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''of a bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c0218401.png" /> in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c0218402.png" />''
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''of a bounded set $M$ in a metric space $(X,\rho)$''
  
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c0218403.png" /> for which
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An element $x_0 \in X$ for which
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\begin{equation}\label{eq:1}
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\sup_{y\in M} \rho(x_0,y) = \inf_{x\in X} \sup_{y\in M} \rho(x,y)
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\end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c0218404.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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The quantity \eqref{eq:1} is the ''[[Chebyshev radius]]'' of the set $M$. If a normed linear space is dual to some normed linear space, then any bounded set $M$ has at least one Chebyshev centre. There exists a Banach space and a three-point set in it that has no Chebyshev centre. Every bounded set in a Banach space $X$ has at most one Chebyshev centre if and only if $X$ is uniformly convex in every direction, that is, if for any $z \in X$ and any $\epsilon > 0$ there exists a number $\delta = \delta(z,\epsilon)>0$ such that if $|| x_1 || = || x_2 || = 1$, $x_1-x_2 = \lambda z$ and $|| x_1 + x_2 || \ge 1-\delta$, then $|\lambda| < \epsilon$. The Chebyshev centre of every bounded set $M$ in a normed linear space $X$ of dimension greater than two is contained in the convex hull of that set if and only if $X$ is a Hilbert space. A Chebyshev centre is a special case of the more general notion of a best $N$-lattice.
  
The quantity (*) is the [[Chebyshev radius|Chebyshev radius]] of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c0218405.png" />. If a normed linear space is dual to some normed linear space, then any bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c0218406.png" /> has at least one Chebyshev centre. There exists a Banach space and a three-point set in it that has no Chebyshev centre. Every bounded set in a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c0218407.png" /> has at most one Chebyshev centre if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c0218408.png" /> is uniformly convex in every direction, that is, if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c0218409.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c02184010.png" /> there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c02184011.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c02184012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c02184013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c02184014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c02184015.png" />. The Chebyshev centre of every bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c02184016.png" /> in a normed linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c02184017.png" /> of dimension greater than two is contained in the convex hull of that set if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c02184018.png" /> is a Hilbert space. A Chebyshev centre is a special case of the more general notion of a best <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021840/c02184019.png" />-lattice.
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====References====
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">A. L. Garkavi, “The theory of best approximation in normed linear spaces”, Itogi Nauki. Ser. Matematika. Mat. Anal. 1967, VINITI, Moscow (1969) 75–132; Progr. Math., '''8''' (1970) 83–150 {{ZBL|0258.41019}}</TD></TR>
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</table>
  
====References====
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{{TEX|done}}
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  ''Itogi Nauki. Mat. Anal. 1967''  (1969)  pp. 75–132</TD></TR></table>
 

Latest revision as of 18:08, 21 February 2018

of a bounded set $M$ in a metric space $(X,\rho)$

An element $x_0 \in X$ for which \begin{equation}\label{eq:1} \sup_{y\in M} \rho(x_0,y) = \inf_{x\in X} \sup_{y\in M} \rho(x,y) \end{equation}

The quantity \eqref{eq:1} is the Chebyshev radius of the set $M$. If a normed linear space is dual to some normed linear space, then any bounded set $M$ has at least one Chebyshev centre. There exists a Banach space and a three-point set in it that has no Chebyshev centre. Every bounded set in a Banach space $X$ has at most one Chebyshev centre if and only if $X$ is uniformly convex in every direction, that is, if for any $z \in X$ and any $\epsilon > 0$ there exists a number $\delta = \delta(z,\epsilon)>0$ such that if $|| x_1 || = || x_2 || = 1$, $x_1-x_2 = \lambda z$ and $|| x_1 + x_2 || \ge 1-\delta$, then $|\lambda| < \epsilon$. The Chebyshev centre of every bounded set $M$ in a normed linear space $X$ of dimension greater than two is contained in the convex hull of that set if and only if $X$ is a Hilbert space. A Chebyshev centre is a special case of the more general notion of a best $N$-lattice.

References

[1] A. L. Garkavi, “The theory of best approximation in normed linear spaces”, Itogi Nauki. Ser. Matematika. Mat. Anal. 1967, VINITI, Moscow (1969) 75–132; Progr. Math., 8 (1970) 83–150 Zbl 0258.41019
How to Cite This Entry:
Chebyshev centre. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_centre&oldid=13013
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article