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This paper is concerned with the various inner and outer radii of a convex bodyC in ad-dimensional normed space. The innerj-radiusrj(C) is the radius of a largestj-ball contained inC, and the outerj-radiusRj(C) measures how wellC can be approximated, in a minimax sense, by a (d —j)-flat. In particular,rd(C) andRd(C) are the usual inradius and circumradius ofC, while 2r1(C) and 2R1(C) areC'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.

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