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 [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

[1]	S.I. [S.I. Zukhovitskii] Zukhovitsky, L.I. Avdeeva, "Linear and convex programming" , Saunders (1966)
[2]	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)
[3]	I.I. Eremin, "Incompatible systems of linear inequalities" Dokl. Akad. Nauk SSSR , 138 : 6 (1961) pp. 1280–1283 (In Russian)

Comments

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

Sometimes Chebyshev is spelled differently as Tschebyshev or Tschebycheff.

References

[a1]	L. Fox, I. Parker, "Chebyshev polynomials in numerical analysis" , Oxford Univ. Press (1968)