Abstract

Floating and Illumination Bodies for Polytopes: Duality Results

Highlights

  • Floating bodies are a fundamental notion in convex geometry

  • Notions of floating bodies are motivated by the physical description of floating objects

  • The purpose of this paper is to make the duality relation between floating body and illumination body precise when the convex body is a polytope P. It was shown by Schütt [19] that the limit of the volume difference of a polytope P and its floating body leads to a quantity related to the combinatorial structure of the polytope, namely the flags of P

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Summary

Introduction

Floating bodies are a fundamental notion in convex geometry. Early notions of floating bodies are motivated by the physical description of floating objects. The purpose of this paper is to make the duality relation between floating body and illumination body precise when the convex body is a polytope P. It was shown by Schütt [19] that the limit of the (appropriately normalized) volume difference of a polytope P and its floating body leads to a quantity related to the combinatorial structure of the polytope, namely the flags of P (see section 5). As in the smooth case [13], a limit procedure leads to a new affine invariant that is not related to the combinatorial structure of the boundary of the polytope, but, as in the smooth case, to cone measures.

Main theorem and consequences
Tools and Lemmas
Upper bound
Lower Bound
The combinatorial structure of dP
Approximation results for the floating body and open questions
Full Text
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