Inner and outer j-radii of convex bodies in finite-dimensional normed spaces

Abstract

This paper is concerned with the various inner and outer radii of a convex body C in a d-dimensional normed space. The inner j-radius rj(C) is the radius of a largest j-ball contained in C, and the outer j-radius Rj(C) measures how well C can be approximated, in a minimax sense, by a (d -j)-flat. In particular, rd(C) and Rd(C) are the usual inradius and circumradius of C, while 2 r1(C) and 2 R1(C) are C's diameter and width. Motivation for the computation of polytope radii has arisen from problems in computer science and mathematical programming. The radii of polytopes are studied in [GK1] and [GK2] from the viewpoint of the theory of computational complexity. This present paper establishes the basic geometric and algebraic properties of radii that are needed in that study. © 1992 Springer-Verlag New York Inc.