Volume ratios for Cartesian products of convex bodies

Khrabrov, A.

doi:10.1090/spmj/1676

Volume ratios for Cartesian products of convex bodies
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by A. I. Khrabrov
Translated by: S. V. Kislyakov

St. Petersburg Math. J. 32 (2021), 905-916

DOI: https://doi.org/10.1090/spmj/1676

Published electronically: August 31, 2021

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Abstract:

The paper is devoted to the behavior of volume ratios, the modified Banach–Mazur distance, and the vertex index for sums of convex bodies. It is shown that \begin{equation*} \sup d (\mathrm {A}\oplus \mathrm {K},\mathrm {B}\oplus \mathrm {L}) \geq \sup \partial (\mathrm {A}\oplus \mathrm {K},\mathrm {B}\oplus \mathrm {L}) \geq c \cdot n^{1-\frac {k+k’}{2n}}, \end{equation*} if $\mathrm {K}\subset \mathbb {R}^n$ and $\mathrm {L}\subset \mathbb {R}^k$ are convex and symmetric (the supremum is taken over all symmetric convex bodies $\mathrm {A}\subset \mathbb {R}^{n-k}$ and $\mathrm {B}\subset \mathbb {R}^{n-k’})$. Furthermore, some examples are discussed that show that the available extimates of the vertex index in terms of the volume ratio are not sharp.

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