# Chebyshev point

of a system of linear inequalities

$$\eta_i(x)=a_{i1}\xi_1+\dots+a_{in}\xi_n+a_i\leq0,\quad i=1,\dots,m,$$

A point $x=(\xi_1,\dots,\xi_n)$ at which the minimax

$$\min_x\max_{1\leq i\leq m}\eta_i(x)$$

is attained. The problem of finding a Chebyshev point reduces to the general problem of linear programming .

A more general notion is that of a Chebyshev point $x^*$ of a system of hyperplanes $\{H_i\}_{i-1}^m$ in a Banach space $X$, i.e. a point $x^*$ for which

$$\sup_{1\leq i\leq m}\inf_{z\in H_i}\|z-x^*\|=\inf_{x\in X}\sup_{1\leq i\leq m}\inf_{z\in H_i}\|z-x\|.$$

Chebyshev points are often chosen as "solutions" of incompatible linear systems of equations and inequalities.

How to Cite This Entry:
Chebyshev point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_point&oldid=43471
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article