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

By A. V. Kolesnikov, Emanuel Milman. 2017

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The L-Brunn–Minkowski theory for p ≥ 1, proposed by Firey and developed by Lutwak in the 90’s, replaces the Minkowski addition of convex sets by its L counterpart, in which the support functions are added in Lnorm. Recently, Böröczky, Lutwak, Yang and Zhang have proposed to extend this theory further to encompass the range p ∈ [0, 1). In particular, they conjectured an L-Brunn–Minkowski inequality for origin-symmetric convex bodies in that range, which constitutes a strengthening of the classical BrunnMinkowski inequality. Our main result confirms this conjecture locally for all (smooth) origin-symmetric convex bodies in R and p ∈ [1 − c n3/2 , 1). In addition, we confirm the local log-Brunn–Minkowski conjecture (the case p = 0) for small-enough C-perturbations of the unit-ball of l q for q ≥ 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 l q with q ∈ [1, 2), we confirm an analogous result for p = c ∈ (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-Minkowski problem, as well as improved stability estimates in the Brunn–Minkowski and anisotropic isoperimetric inequalities. Faculty of Mathematics, Higher School of Economics, Moscow, Russia. The author was supported by RFBR project 17-01-00662 and DFG project RO 1195/12-1. The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2017–2018 (grant No 17-01-0102) and by the Russian Academic Excellence Project “5-100”. Emails: akolesnikov@hse.ru, sascha77@mail.ru. Department of Mathematics, Technion Israel Institute of Technology, Haifa 32000, Israel. The research leading to these results is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 637851). Email: emilman@tx.technion.ac.il. 2010 Mathematics Subject Classification: 52A40, 52A23, 35P15, 58J50.

Subfields: Applied Mathematics, Pharmacology

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