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

Abstract

The L p L^p -Brunn–Minkowski theory for p ≥ 1 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 L^p counterpart, in which the support functions are added in L p L^p -norm. Recently, Böröczky, Lutwak, Yang and Zhang have proposed to extend this theory further to encompass the range p ∈ [ 0 , 1 ) p \in [0,1) . In particular, they conjectured an L p 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 R n \mathbb {R}^n and p ∈ [ 1 − c n 3 / 2 , 1 ) p \in [1 - \frac {c}{n^{3/2}},1) . In addition, we confirm the local log-Brunn–Minkowski conjecture (the case p = 0 p=0 ) for small-enough C 2 C^2 -perturbations of the unit-ball of ℓ q n \ell _q^n for q ≥ 2 q \geq 2 , when the dimension n n is sufficiently large, as well as for the cube, which we show is the conjectural extremal case. For unit-balls of ℓ q n \ell _q^n with q ∈ [ 1 , 2 ) q \in [1,2) , we confirm an analogous result for p = c ∈ ( 0 , 1 ) 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 L^p -Minkowski problem, as well as improved stability estimates in the Brunn–Minkowski and anisotropic isoperimetric inequalities.