Floating Body, Illumination Body, and Polytopal Approximation

Summary

Let K be a convex body in IRd and K t its floating bodies. There is a polytope that satisfies K_t ⊂ P_n ⊂ K and has at most n vertices, where

Let K^t be the illumination bodies of K and Q_n a polytope that contains K and has at most n (d-1)-dimensional faces.

1. Introduction

We investigate the approximation of a convex body K in Rd by a polytope. We measure the approximation by the symmetric difference metric. The symmetric difference metric between two convex bodies K and C is

We study in particular two questions: How well can a convex body K be approximated by a polytope P_n that is contained in K and has at most n vertices and how well can K be approximated by a polytope Q_n that contains K and has at most n (d-1)-dimensional faces. Macbeath [Mac] showed that the Euclidean Ball B^d₂ is an extremal case: The approximation for any other convex body is better. We have for the Euclidean ballprovided that n ≥ (C₃ d)^(d-1)/2. The right hand inequality was first established by Bronshtein and Ivanov [BI] and Dudley [D₁,D₂]. Gordon, Meyer, and Reisner [GMR₁,GMR₂] gave a constructive proof for the same inequality. Muller [Mü] showed that random approximation gives the same estimate. Gordon, Reisner, and Schutt [GRS] established the left hand inequality. Gruber [Gr₂] obtained an asymptotic formula.