Local Lp-Brunn-Minkowski inequalities for p<1
The Lp-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 Lp counterpart, in which the support functions are added in Lp-norm. 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 Lp-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 Rn and p∈[1−cn3/2,1). In addition, we confirm the local log-Brunn--Minkowski conjecture (the case p=0) for small-enough C2-perturbations of the unit-ball of ℓnq 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 ℓnq 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 Lp-Minkowski problem, as well as improved stability estimates in the Brunn-Minkowski and anisotropic isoperimetric inequalities.