Duality of Floating and Illumination Bodies

TL;DR

This paper explores a duality relationship between floating and illumination bodies in convex geometry, providing estimates that show equality only for ellipsoids, thus deepening understanding of their geometric properties.

Contribution

It establishes precise estimates for the duality relation in centrally symmetric convex bodies, highlighting the unique case of ellipsoids where equality holds.

Findings

Equality of the polar of the floating body and the illumination body of the polar only for ellipsoids.

Provides estimates for the class of $B_p^n$ and convex bodies with positive Gauss curvature.

Deepens understanding of duality relations in convex geometric bodies.

Abstract

We investigate a duality relation between floating and illumination bodies. The definitions of these two bodies suggest that the polar of the floating body should be similar to the illumination body of the polar. We consider this question for the class of centrally symmetric convex bodies. We provide precise estimates for $(B_p^n)$ and for centrally symmetric convex bodies with everywhere positive Gauss curvature. Our estimates show that equality of the polar of the floating body and the illumination body of the polar can only be achieved in the case of ellipsoids.