Local L^p-Brunn-Minkowski Inequalities for p < 1

The L^p-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^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 ∈ [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 Rⁿ and (Equation presented), 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 ιⁿ_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 ιⁿ_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^p-Minkowski problem, as well as improved stability estimates in the Brunn-Minkowski and anisotropic isoperimetric inequalities.