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
Summary
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.
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