This paper is concerned with convex bodies in n-dimensional lp spaces, where each body is accessible only by a weak separation or optimization oracle. It studies the asymptotic relative accuracy, as n → ∞, of polynomial-time approximation algorithms for the diameter, width, circumradius, and inradius of a body K, and also for the maximum of the norm over K. In the case of l2 (Euclidean n-space), a 1987 result of Bárány and Füredi severely limits the degree of relative accuracy that can be guaranteed in approximating K's volume by any deterministic polynomial-time algorithm. This led to a similarly severe limit on the relative accuracy of deterministic polynomial-time algorithms for computing K's diameter. However, these limitations on the accuracy of deterministic computation were soon followed by the work of Dyer, Frieze and Kannan showing that, for volume approximation, arbitrarily good accuracy can be attained with the aid of suitable randomization. It was therefore natural to wonder whether the same is true of the diameter. The first main result of this paper is that, in contrast to the situation for the volume, randomization does not help in approximating the diameter. The same limitation on accuracy that applies to deterministic polynomial-time computation still applies when randomization is permitted. This conclusion applies also to the width, circumradius, and inradius of a body, and to maximization of the norm over K. The second main result is that, for each of the five "radius" measurements just mentioned, the inapproximability results for deterministic polynomial-time approximation are optimal for width and inradius when 1≤p≤2, are optimal for diameter, circumradius, and norm-maximization when 2≤p≤∞, and in the remaining cases are within a logarithmic factor of being optimal. In particular, all are optimal when p = 2. The optimality is established by producing deterministic polynomial-time approximation algorithms whose accuracy is bounded below by a positive constant multiple (independent of the dimension n) of the upper bounds on accuracy. Since the bodies are assumed to be presented by a weak oracle, our approach belongs to the algorithmic theory of convex bodies initiated by Grötschel, Lovász and Schrijver. In the deterministic case we sharpen and extend l2 results due to these authors, and in the randomized case we refine some ideas presented earlier by Lovász and Simonovits. The algorithms that establish lower bounds on accuracy use certain polytopal approximations of lp unit balls that are obtained by polarizing and extending an l2 method of Kochol. The argumentation for upper bounds requires, in addition to extending the l2 approach of Bárány and Füredi, a careful treatment of some results on entropy numbers used by Carl and Pajor. It is closely related to some questions concerning sphere coverings.
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