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Local $L^p$-Brunn–Minkowski inequalities for $p < 1$

About this Title

Alexander V. Kolesnikov and Emanuel Milman

Publication: Memoirs of the American Mathematical Society
Publication Year: 2022; Volume 277, Number 1360
ISBNs: 978-1-4704-5160-8 (print); 978-1-4704-7092-0 (online)
DOI: https://doi.org/10.1090/memo/1360
Published electronically: March 25, 2022
Keywords: $L^p$-Brunn–Minkowski theory, convex bodies, Aleksandrov body, Hilbert–Brunn–Minkowski operator, Poincaré inequality, local uniqueness in $L^p$-Minkowski problem, isoperimetric stability estimates

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Notation
  • 3. Global vs. Local Formulations of the $L^p$-Brunn–Minkowski Conjecture
  • 4. Local $L^p$-Brunn–Minkowski Conjecture – Infinitesimal Formulation
  • 5. Relation to Hilbert–Brunn–Minkowski Operator and Linear Equivariance
  • 6. Obtaining Estimates via the Reilly Formula
  • 7. The second Steklov operator and $B_H(B_2^n)$
  • 8. Unconditional Convex Bodies and the Cube
  • 9. Local log-Brunn–Minkowski via the Reilly Formula
  • 10. Continuity of $B_H$, $B$, $D$ with application to $B_q^n$
  • 11. Local Uniqueness for Even $L^p$-Minkowski Problem
  • 12. Stability Estimates for Brunn–Minkowski and Isoperimetric Inequalities

Abstract

The $L^p$-Brunn–Minkowski theory for $p\geq 1$, proposed by Firey and developed by Lutwak in the 90’s, replaces the Minkowski addition of convex sets by its $L^p$ counterpart, in which the support functions are added in $L^p$-norm. Recently, Böröczky, Lutwak, Yang and Zhang have proposed to extend this theory further to encompass the range $p \in [0,1)$. In particular, they conjectured an $L^p$-Brunn–Minkowski inequality for origin-symmetric convex bodies in that range, which constitutes a strengthening of the classical Brunn-Minkowski inequality. Our main result confirms this conjecture locally for all (smooth) origin-symmetric convex bodies in $\mathbb {R}^n$ and $p \in [1 - \frac {c}{n^{3/2}},1)$. In addition, we confirm the local log-Brunn–Minkowski conjecture (the case $p=0$) for small-enough $C^2$-perturbations of the unit-ball of $\ell _q^n$ for $q \geq 2$, when the dimension $n$ is sufficiently large, as well as for the cube, which we show is the conjectural extremal case. For unit-balls of $\ell _q^n$ with $q \in [1,2)$, we confirm an analogous result for $p=c \in (0,1)$, a universal constant. It turns out that the local version of these conjectures is equivalent to a minimization problem for a spectral-gap parameter associated with a certain differential operator, introduced by Hilbert (under different normalization) in his proof of the Brunn–Minkowski inequality. As applications, we obtain local uniqueness results in the even $L^p$-Minkowski problem, as well as improved stability estimates in the Brunn–Minkowski and anisotropic isoperimetric inequalities.

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