On a quantitative reversal of Alexandrov’s inequality
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- by Grigoris Paouris, Peter Pivovarov and Petros Valettas PDF
- Trans. Amer. Math. Soc. 371 (2019), 3309-3324 Request permission
Abstract:
Alexandrov’s inequalities imply that for any convex body $A$, the sequence of intrinsic volumes $V_1(A),\ldots ,V_n(A)$ is non-increasing (when suitably normalized). Milman’s random version of Dvoretzky’s theorem shows that a large initial segment of this sequence is essentially constant, up to a critical parameter called the Dvoretzky number. We show that this near-constant behavior actually extends further, up to a different parameter associated with $A$. This yields a new quantitative reverse inequality that sits between the approximate reverse Urysohn inequality, due to Figiel–Tomczak–Jaegermann and Pisier, and the sharp reverse Urysohn inequality for zonoids, due to Hug–Schneider. In fact, we study concentration properties of the volume radius and mean width of random projections of $A$ and show how these lead naturally to such reversals.References
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Additional Information
- Grigoris Paouris
- Affiliation: Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 671202
- Email: grigoris@math.tamu.edu
- Peter Pivovarov
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 824727
- Email: pivovarovp@missouri.edu
- Petros Valettas
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 957443
- Email: valettasp@missouri.edu
- Received by editor(s): February 19, 2017
- Received by editor(s) in revised form: August 23, 2017
- Published electronically: December 7, 2018
- Additional Notes: The first author was supported by NSF grant CAREER-1151711.
The second and third authors were supported by NSF grant DMS-1612936. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 3309-3324
- MSC (2010): Primary 52A23; Secondary 52A39, 52A40
- DOI: https://doi.org/10.1090/tran/7413
- MathSciNet review: 3896113