Polytopal approximation of elongated convex bodies

TL;DR

This paper investigates how well elongated convex bodies can be approximated by polytopes with a fixed number of facets, providing bounds based on the body's elongation measured by an isoperimetric ratio.

Contribution

It introduces bounds for the Hausdorff approximation of elongated convex bodies by circumscribed polytopes, emphasizing the role of elongation measured via an isoperimetric ratio.

Findings

Bounds depend on the body's elongation

Approximation quality improves with more facets

Elongation significantly affects approximation bounds

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.