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

Abstract

The $L^p$-Brunn-Minkowski theory for $p\ge 1$, 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 $q\ge 2$, 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.

85 pages; corrected typos, and added a section with additional applications regarding new and improved stability estimates in the Brunn-Minkowski and anisotropic isoperimetric inequalities

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Cited by in corpus (8)

• On the $L_p$ Brunn-Minkowski theory and the $L_p$ Minkowski problem for $C$-coconvex sets
• Wulff shapes and a characterization of simplices via a Bezout type inequality
• Equivalence of the local and global versions of the $L^p$-Brunn-Minkowski inequality
• Stability of the log-Brunn-Minkowski inequality in the case of many hyperplane symmetries
• $L^p$-Minkowski Problem under Curvature Pinching
• $L_p$-Brunn-Minkowski inequality for $p\in (1-\frac{c}{n^{\frac{3}{2}}},1)$
• Stability of the Prekopa-Leindler inequality for log-concave functions
• Stable solution of the log-Minkowski problem in the case of many hyperplane symmetries