Volume ratios for Cartesian products of convex bodies
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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.References
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Bibliographic Information
- A. I. Khrabrov
- Affiliation: HSE University, St. Petersburg, St. Petersburg School of Mathematics Physics and Computer Science, 194100, ul. Kantemirovskaya, 3 building 1, lit. A, St. Petersburg, Russia –and– St. Petersburg State University, Department of Mathematics and Computer Science, St. Petersburg, 199178, Line 14, 29B, V.O., St. Petersburg, Russia
- Email: aikhrabrov@mail.ru
- Received by editor(s): November 3, 2019
- Published electronically: August 31, 2021
- © Copyright 2021 American Mathematical Society
- Journal: St. Petersburg Math. J. 32 (2021), 905-916
- MSC (2020): Primary 52A21
- DOI: https://doi.org/10.1090/spmj/1676
- MathSciNet review: 4167875
Dedicated: To the bright memory of my teacher, Boris Mikhailovich Makarov