Uniform Estimation of a Convex Body by a Fixed-Radius Ball

Abstract

The paper deals with a finite-dimensional problem of finding a uniform estimate (approximation in the Hausdorff metric) of a convex body by a fixed-radius ball in an arbitrary norm. The solution of the problem and its properties essentially depends on the radius value. We used the solutions of simpler problems of circumscribed and inscribed balls for the given convex body to divide the positive radius scale into ranges. Taking into account these ranges, we derived necessary and sufficient conditions for the solutions of the problem by means of convex analysis including properties of distant functions (to the nearest and most distant points of the set) and their differential characteristics. Also, under additional assumptions concerning the estimated convex body or the used norm, conditions of solution uniqueness and variational properties of the solution were obtained for various ranges of radius values.