Computational complexity of inner and outer j-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 an n-dimensional convex polytope in the space ∝ⁿ equipped with an ℓ_p norm or a polytopal norm. The polytope P 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 (ℋ-presented). The inner j-radius of P is the radius of a largest j-ball contained in P; it is P's inradius when j = n and half of P's diameter when j = 1. The outer j-radius measures how well P can be approximated, in a minimax sense, by an (n - j)-flat; it is P's circumradius when j = n and half of P's width when j = 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 which P is centrally symmetric. When the dimension n is permitted to vary, the situation is roughly as follows: (a) for general ℋ-presented polytopes in ℓ_p spaces with 1<p<∞, all outer radius computations are NP-hard; (b) in the remaining cases (including symmetric ℋ-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.