Floating and Illumination Bodies for Polytopes: Duality Results

TL;DR

This paper investigates the relationship between floating bodies and illumination bodies of polytopes, establishing convergence results and introducing a new affine invariant linked to the cone measure.

Contribution

It provides the first precise convergence results for centrally symmetric polytopes and introduces a novel affine invariant related to the cone measure.

Findings

Established convergence results for centrally symmetric polytopes.

Introduced a new affine invariant connected to the cone measure.

Enhanced understanding of duality between floating and illumination bodies.

Abstract

We consider the question how well a floating body can be approximated by the polar of the illumination body of the polar. We establish precise convergence results in the case of centrally symmetric polytopes. This leads to a new affine invariant which is related to the cone measure of the polytope.