Chebyshev centre

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