On volume product inequalities for convex sets

Campi, Stefano; Gronchi, Paolo

doi:10.1090/S0002-9939-06-08241-4

On volume product inequalities for convex sets
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by Stefano Campi and Paolo Gronchi

Proc. Amer. Math. Soc. 134 (2006), 2393-2402

DOI: https://doi.org/10.1090/S0002-9939-06-08241-4

Published electronically: February 3, 2006

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Abstract:

The volume of the polar body of a symmetric convex set $K$ of ${\mathbb {R}^d}$ is investigated. It is shown that its reciprocal is a convex function of the time $t$ along movements, in which every point of $K$ moves with constant speed parallel to a fixed direction. This result is applied to find reverse forms of the $L^{p}$-Blaschke-Santaló inequality for two-dimensional convex sets.

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