Asymptotic approximation of smooth convex bodies by polytopes

We study asymptotic properties of the approximation of a sufficiently smooth convex body Km (R by the convex hulls of n points in the boundary of K, for n -» oo. The deviation is measured by the Hausdorff distance. The asymptotic distribution of the vertices of bestapproximating polytopes is determined. Further results involve prescribed densities for the vertices and describe the strength of approximation by either deterministic or random polytopes. 1991 Mathematics Subject Classification: 41A60, 52A20, 52A27, 60D05; 41A50, 53B21. l Introduction and Statement of results In a well-known paper of 1975, McClure and Vitale [11] studied the approximation of sufficiently smooth convex curves in the plane by inscribed or circumscribed polygons with n vertices, for n -> oo. With respect to different measures of deviation, they obtained sharp estimates of the order of convergence of best approximations, asymptotic characterizations of best approximations, and methods for the construction of asymptotically efficient approximations. It is a challenge to extend their results to higher dimensions, where different methods are required. A few of the results of [l 1] have been generalized in this way. We refer to Gruber [7] for a recent survey over approximation problems for convex bodies. In the present paper, we continue the line of research begun by McClure and Vitale [11] and extend two more of their theorems to sufficiently smooth convex bodies in d-dimensional space. Also a related result from [16] on random approximation will be generalized in a similar way. We restrict our considerations to approximation by inscribed polytopes and to measuring the deviation in terms of the Hausdorff metric; some hints to further possibilities will be given in Section 5. 364 S. Glasauer, R. Schneider By Jf we denote the set of convex bodies (non-empty, compact, convex subsets) in d-dimensional Euclidean space (R (d > 2). Here iR is equipped with the Standard scalar product and induced norm, and Jf carries the Hausdorff metric, denoted by δ. (For notions and results from the theory of convex bodies that are used without explanation, we refer to [17].) A convex body Ke tf is said to be of class C+ , where k>2, if K has interior points and its boundary, denoted by bdK, is a regul r hypersurface of differentiability class C with everywhere positive curvatures. The Gauss-Kronecker curvature of K at χ e bdK is denoted by κκ(χ). For K E Jf and w e N, we denote by &n(K) the set of convex polytopes contained in K and having at most n vertices. There exists a polytope P„*, not unique in general, for which We assume, without loss of generality, that P* has all its vertices on the boundary of K. In the following, the constants Kk and 5fc are, respectively, the volume of the >dimensional unit ball and the minimum density of coverings of fR by unit balls. By σ we denote the Euclidean surface area measure on bdK. We first recall an older result. Theorem 1. I f K e J f d is α convex body of class C+, then 2/(d-i> Λ -3^ / Ο Γ Λ + Ο Ο . (1) d-l bdK Theorem l was proved in [l 3] for bodies of class C+ and was obtained by Gruber [5] under the C+ assumption. It extends a result of L. Fejes Toth [3] and Theorem 5 (i) of McClure and Vitale [11] from two to d dimensions. Theorem l describes the asymptotic order of approximation by the best-approximating polytopes /J*. While there is little hope to characterize the polytopes P* individually, the asymptotic distribution of their vertices can be described precisely. This is done in the next theorem, which extends Theorem 6 (i) of McClure and Vitale [11]. For its formulation, it is convenient to use some terminology from the theory of uniform distribution of sequences. In particular, we combine the notions of uniform distribution of double sequences (see Hlawka [8], p. 57) and of uniform distribution with respect to a finite measure (Kuipers and Niederreiter [10], Chapter 3) in a suitable way. Let (5Π)Λ6Ν be a sequence of finite subsets of bd^ with cardSn -» oo for n ~* oo, and let h : bdK -» IR* be a positive continuous function on bdK. The sequence (S„)n6N is called uniformly distributed with density h if