Shadow systems and volume of polar convex bodies

TL;DR

This paper proves convexity properties of the reciprocal volume of polar bodies in shadow systems, extending known results to non-symmetric cases and applying findings to reverse Santaló inequalities for specific polytopes.

Contribution

It extends the convexity of reciprocal volume results to non-symmetric shadow systems and characterizes cases where this reciprocal is affine, with applications to reverse Santaló inequalities.

Findings

Reciprocal volume of polar bodies in shadow systems is convex in parameter t.

Characterization of when the reciprocal volume is affine in t.

Application to reverse Santaló inequality for polytopes with up to d+3 vertices.

Abstract

We prove that the reciprocal of the volume of the polar bodies, about the Santal\'o point, of a {\em shadow system} of convex bodies $K_t$, is a convex function of $t$. Thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of $t$ is also investigated and is characterized under certain conditions. We apply these results to prove exact reverse Santal\'o inequality for polytopes in $\rd{d}$ that have at most $d+3$ vertices.