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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Sharp isoperimetric inequalities for affine quermassintegrals
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by Emanuel Milman and Amir Yehudayoff
J. Amer. Math. Soc. 36 (2023), 1061-1101
DOI: https://doi.org/10.1090/jams/1013
Published electronically: December 12, 2022

Abstract:

The affine quermassintegrals associated to a convex body in $\mathbb {R}^n$ are affine-invariant analogues of the classical intrinsic volumes from the Brunn–Minkowski theory, and thus constitute a central pillar of Affine Convex Geometry. They were introduced in the 1980’s by E. Lutwak, who conjectured that among all convex bodies of a given volume, the $k$-th affine quermassintegral is minimized precisely on the family of ellipsoids. The known cases $k=1$ and $k=n-1$ correspond to the classical Blaschke–Santaló and Petty projection inequalities, respectively. In this work we confirm Lutwak’s conjecture, including characterization of the equality cases, for all values of $k=1,\ldots ,n-1$, in a single unified framework. In fact, it turns out that ellipsoids are the only local minimizers with respect to the Hausdorff topology.

For the proof, we introduce a number of new ingredients, including a novel construction of the Projection Rolodex of a convex body. In particular, from this new view point, Petty’s inequality is interpreted as an integrated form of a generalized Blaschke–Santaló inequality for a new family of polar bodies encoded by the Projection Rolodex. We extend these results to more general $L^p$-moment quermassintegrals, and interpret the case $p=0$ as a sharp averaged Loomis–Whitney isoperimetric inequality.

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Bibliographic Information
  • Emanuel Milman
  • Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
  • MR Author ID: 696280
  • Email: emilman@tx.technion.ac.il
  • Amir Yehudayoff
  • Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
  • MR Author ID: 815742
  • Email: amir.yehudayoff@gmail.com
  • Received by editor(s): December 17, 2020
  • Received by editor(s) in revised form: June 14, 2022
  • Published electronically: December 12, 2022
  • Additional Notes: The research leading to these results was part of a project that had received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 637851). This work was supported by the Israeli Science Foundation grant #1162/15.
  • © Copyright 2022 by the authors.
  • Journal: J. Amer. Math. Soc. 36 (2023), 1061-1101
  • MSC (2020): Primary 52A40
  • DOI: https://doi.org/10.1090/jams/1013
  • MathSciNet review: 4618955