Peter M. Gruber

Aspects of Approximation of Convex Bodies

Publisher Summary This chapter reviews various aspects of approximation of convex bodies. Approximation of convex bodies is frequently encountered in geometric convexity, discrete geometry, the theory of finite-dimensional normed spaces, in geometric algorithms and optimization, and in the realm of engineering. Also, approximation problems in optimization arise often from more practical problems of operations research and pattern recognition. Several effective approximation algorithms formulated for convex functions or convex bodies are described in the chapter. In the former case the approximation is considered with respect to the maximum norm, in the latter case with respect to the Hausdorff metric. The chapter presents more recent developments in approximation, but many older results are also described.

Approximation of convex bodies

Approximation of convex bodies by either smooth convex bodies or polytopes has been considered frequently in convexity. There are two incentives for it. On the one hand approximation is used as a tool for investigations ranging from classical results on mixed volumes to questions concerning the e-entropy of spaces of convex bodies, on the other hand there is an intrinsic geometric interest in the approximation problem itself.

Asymptotic estimates for best and stepwise approximation of convex bodies III

We consider approximations of a smooth convex body by inscribed and circumscribed convex polytopes as the number of vertices, resp. facets tends to infinity. The measure of deviation used is the dierence of the mean width of the convex body and the approximating polytopes. The following results are obtained. (i) An asymptotic formula for best approximation. (ii) Upper and lower estimates for step- by-step approximation in terms of the so-called dispersion. (iii) For a sequence of best approximating inscribed polytopes the sequence of vertex sets is uniformly distributed in the boundary of the convex body where the density is specified explicitly.

The Space of Convex Bodies

This chapter discusses the space of convex bodies. A convex body in E d is a compact convex subset of E d . It is called proper if it has non-empty interior. The spaces E d (the space of all convex bodies in E d ) and E p (the subspace of all proper convex bodies) and some of their subspaces have been investigated for many decades, an early result being Blaschkes selection theorem. E d is boundedly complete and k p is locally compact. The proof published by Blaschke follows an idea of Caratheodory who suggested the use of ɛ-nets. Blaschkes original proof was based on a result of Hilbert which was a forerunner of the theorem of Arzela–Ascoli. Boi and Heil proved that each of the results of Blaschke and Arzela–Ascoli is a consequence of the other one. K being a locally compact space there are many Borel measures on it. Lattices of convex bodies have been studied mainly by Belgian mathematicians. In most of their results, E d is replaced by topological vector spaces of arbitrary dimensions.

Volume approximation of convex bodies by inscribed polytopes

A convex body C in d-dimensional Euclidean space E ~ is a compact convex subset of lE a with non-empty interior. We say that C is of differentiability class ~k if its boundary considered as a manifold is of class ~k. The symmetric difference metric 6 s, also called Nikodym metric, on the space of convex bodies is defined for any two convex bodies as the volume of their symmetric difference. Given a convex body C, let ~ = ~ ( C ) (n = d + l, d + 2,...) denote the family of convex polytopes P having at most n vertices and being inscribed into C, that is, all vertices are on the boundary bdC of C. It is well known that for each n there is a polytope P, ~ ~ such that ~s(C, P.) = ~s(C, ~,~)( = inf {~s(C, P): P ~ ~.~}).

Approximation of convex bodies by polytopes

LetC be a convex body ofEd and consider the symmetric difference metric. The distance ofC to its best approximating polytope having at mostn vertices is 0 (1/n2/(d−1)) asn→∞. It is shown that this estimate cannot be improved for anyC of differentiability class two. These results complement analogous theorems for the Hausdorff metric. It is also shown that for both metrics the approximation properties of «most» convex bodies are rather irregular and that ford=2 «most» convex bodies have unique best approximating polygons with respect to both metrics.

Problems in Geometric Convexity

The following list consists of problems collected at the Oberwolfach conferences on Convexity in 1974, 1976, 1978. We have reproduced each of these problems, even if it has been solved in the meantime. Recent information about problems and references to solutions or partial solutions that has come to our knowledge has been added at appropriate places.

Geometry of the cone of positive quadratic forms

Abstract Let a quadratic form on 𝔼 d be represented by its coefficient vector in 𝔼(1/2)d(d+1). Then, to the family of all positive semidefinite quadratic forms on 𝔼 d there corresponds a closed convex cone 𝒬 d in 𝔼(1/2)d(d+1) with apex at the origin. We describe its exposed faces and show that the families of its extreme and exposed faces coincide. Using these results, flag transitivity, neighborliness, singularity and duality properties of 𝒬 d are shown. The isometries of the cone 𝒬 d are characterized and we state a conjecture describing its linear symmetries. While the cone 𝒬 d is far from being polyhedral, the results obtained show that it shares many properties with highly symmetric, neighborly and self dual polyhedral convex cones.

Approximation by Convex Polytopes

After some general introductory remarks on approximation in convex geometry we present in the main part of this article asymptotic results on best approximation of convex bodies as the number of vertices, resp. facets of the approximating poly topes tends to infinity. Tools are from (affine) differential geometry. Since the transparent geometric situation in the plane admits much more precise results, this case is treated separately. In higher dimensions the relations to Dirichlet- Voronoi and Delone tilings and to the ball covering problem are indicated. Then algorithmic and, in particular, step-by-step approximation results are discussed. Here tools from number theory are applied. In the material presented emphasis is on the ideas underlying the proofs. Supplementing earlier surveys, the last chapter contains a summary of recent results on approximation of convex bodies.

Claude Ambrose Rogers. 1 November 1920 — 5 December 2005

Claude Ambrose Rogers and his identical twin brother, Stephen Clifford, were born in Cambridge in 1920 and came from a long scientific heritage. Their great-great-grandfather, Davies Gilbert, was President of the Royal Society from 1827 to 1830; their father was a Fellow of the Society and distinguished for his work in tropical medicine. After attending boarding school at Berkhamsted with his twin brother from the age of 8 years, Ambrose, who had developed very different scientific interests from those of his father, entered University College London in 1938 to study mathematics. He completed the course in 1940 and graduated in 1941 with first-class honours, by which time the UK had been at war with Germany for two years. He joined the Applied Ballistics Branch of the Ministry of Supply in 1940, where he worked until 1945, apparently on calculations using radar data to direct anti-aircraft fire. However, this did not lead to research interests in applied mathematics, but rather to several areas of pure mathematics. Ambroses PhD research was at Birkbeck College, London, under the supervision of L. S. Bosanquet and R. G. Cooke. Although his first paper was a short note on linear transformations of convergent series, his substantive early work was on the geometry of numbers. Later, Rogers became known for his very wide interests in mathematics, including not only geometry of numbers but also Hausdorff measures, convexity and analytic sets, as described in this memoir. Ambrose was married in 1952 to Joan North, and they had two daughters, Jane and Petra, to form a happy family.