Polytopal approximation of elongated convex bodies

Abstract

This paper presents bounds for the best approximation, with respect to the Hausdorff metric, of a convex body K by a circumscribed polytope P with a given number of facets. These bounds are of particular interest if K is elongated. To measure the elongation of the convex set, its isoperimetric ratio V j (K) 1/j V i (K) −1/i is used.

Cells with many facets in a Poisson hyperplane tessellation

“…Proof. This is a useful application of a recent result by Bonnet [1]. Assume 1 ≤ i < j ≤ ⌈(d − 1)/2⌉.…”

“…The starting point of the following considerations is Theorem 1.1 of [1]. For 1 ≤ i < j ≤ d and ǫ > 0, we say that a convex body K is (ǫ : i, j)-elongated when V j (K) 1/j V i (K) −1/i < ǫ.…”

“…Lemma 3.5. Let 1 ≤ i < j ≤ d. There exists a constant c 13 such that for any δ ∈ (0, 1) and for any K, L ∈ K with K ⊂ L and d H (K, L) < δV 1 (K), we have 1) .…”