View source for Chebyshev point

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''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 [[#References|[1]]].

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.I. [S.I. Zukhovitskii] Zukhovitsky,   L.I. Avdeeva,   "Linear and convex programming" , Saunders  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. Belobrov,   "The Chebyshev point of a system of translates of subspaces in a Banach space"  ''Mat. Zametki'' , '''8''' :  4  (1970)  pp. 29–40  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. Eremin,   "Incompatible systems of linear inequalities"  ''Dokl. Akad. Nauk SSSR'' , '''138''' :  6  (1961)  pp. 1280–1283  (In Russian)</TD></TR></table>



====Comments====
The term  "Chebyshev point"  or  "Chebyshev node"  is also used to denote a zero of a Chebyshev polynomial (cf. [[Chebyshev polynomials|Chebyshev polynomials]]) in the theory of (numerical) interpolation, integration, etc. [[#References|[a1]]].

Sometimes Chebyshev is spelled differently as Tschebyshev or Tschebycheff.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Fox,   I. Parker,   "Chebyshev polynomials in numerical analysis" , Oxford Univ. Press  (1968)</TD></TR></table>