Computational complexity of inner and outerj-radii of polytopes in finite-dimensional normed spaces

This paper studies the complexity of computing (or approximating, or bounding) the various inner and outer radii of ann-dimensional convex polytope in the space ⁿ equipped with an _p norm or a polytopal norm. The polytopeP is assumed to be presented as the convex hull of finitely many points with rational coordinates (V-presented) or as the intersection of finitely many closed halfspaces defined by linear inequalities with rational coefficients (-presented). The innerj-radius ofP is the radius of a largestj-ball contained inP; it isP's inradius whenj = n and half ofP's diameter whenj = 1. The outerj-radius measures how wellP can be approximated, in a minimax sense, by an (n — j)-flat; it isP's circumradius whenj = n and half ofP's width whenj = 1. The binary (Turing machine) model of computation is employed. The primary concern is not with finding optimal algorithms, but with establishing polynomial-time computability or NP-hardness. Special attention is paid to the case in whichP is centrally symmetric. When the dimensionn is permitted to vary, the situation is roughly as follows: (a) for general -presented polytopes in _p spaces with 1, all outer radius computations are NP-hard; (b) in the remaining cases (including symmetric -presented polytopes), some radius computations can be accomplished in polynomial time and others are NP-hard. These results are obtained by using a variety of tools from the geometry of convex bodies, from linear and nonlinear programming, and from the theory of computational complexity. Applications of the results to various problems in mathematical programming, computer science and other fields are included.Many of the results of this paper and also of [5, 14–17] were obtained in 1987 while both authors were visiting the Institute for Mathematics and its Applications, Minneapolis, MN 55455. The papers [5, 14–16] have appeared as IMA preprints.Research supported in part by the Alexander-von-Humboldt foundation and by the Deutsche Forschungsgemeinschaft.Research supported in part by the National Science Foundation.