Chebyshev centres, Jung constants, and their applications

Abstract

The approximation of concrete function classes is the most common subject in the theory of approximations of functions. An important particular case of this is the problem of the Chebyshev centre and radius. As it turns out, this problem is not only a special case of the Kolmogorov width problem, but it is also related in a mysterious way to other important characteristics and results in the theory of functions and other more general branches of analysis and geometry. The aim of the present study is to give a survey of the current state of this problem and to discuss its possible applications. Bibliography: 169 titles.

The extreme polygons for the self Chebyshev radius of the boundary

“…It is also proved in [19] that all C 2 -smooth convex curves have good approximations by polygonal chains in the sense of the self Chebyshev radius (1). This observation leads to natural extremal problems for convex polygons.…”

“…For a given point A ∈ Γ we have d(A, A i ) = d(A, A j ) if and only if A is on the straight line trough the point 1 2 (A i + A j ) perpendicular to [A i , A j ]. Hence, we have only finite numbers of points A ∈ Γ such that the set F (A) has more that one point.…”

“…Moreover, this maximum perimeter coincides with the perimeter of the regular 2ngon of diameter d, as E. Makai, J. noticed (a private conversation). Indeed, if P n is such a n-gon, then the diameter of the polygon 1 2 (P n − P n ) (this is the central symmetral of P n , that have the same perimeter as P n has) is also equal to d, 1 2 (P n − P n ) is inscribed into a circle of diameter d, and has ≤ 2n sides. So, this perimeter is no more than the perimeter of the regular 2n-gon incribed into this circle (that can obtained via the above procedure from a regular n-gon of diameter d).…”

On property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps in $\ell_{\infty}$-direct sums

“…Hence py 1 , y 2 q P cent Y pB, δq. Then there exists y 1 1 P cent Y 1 pB 1 q and y 1 2 P cent Y 2 pB 2 q such that }py 1 , y 2 q ´py 1 1 , y 1 2 q} ď ε. Thus }y 1 ´y1 1 } ď ε.…”

Some stability properties of restricted Chebyshev centers in Banach spaces

“…This is evident through [2]- [4], [6], [9], [10], [12], [14] and the references therein. One can also refer [1], which is a recent survey article discussing the current state of few parts of this study.…”