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Shadow systems and volume of polar convex bodies

Mathieu Meyer·Shlomo Reisner
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Abstract

We prove that the reciprocal of the volume of the polar bodies, about the Santaló 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 Santaló inequality for polytopes in $\rd{d}$ that have at most $d+3$ vertices.

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Mathieu Meyer, Shlomo Reisner. 2008-07-14. Shadow systems and volume of polar convex bodies. https://arxiv.org/abs/math/0606305

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