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The $L^p$-Brunn-Minkowski theory for $p\geq 1$, proposed by Firey anddeveloped by Lutwak in the 90's, replaces the Minkowski addition of convex setsby its $L^p$ counterpart, in which the support functions are added in$L^p$-norm. Recently, B\"{o}r\"{o}czky, Lutwak, Yang and Zhang have proposed toextend this theory further to encompass the range $p \in [0,1)$. In particular,they conjectured an $L^p$-Brunn-Minkowski inequality for origin-symmetricconvex bodies in that range, which constitutes a strengthening of the classicalBrunn-Minkowski inequality. Our main result confirms this conjecture locallyfor 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--Minkowskiconjecture (the case $p=0$) for small-enough $C^2$-perturbations of theunit-ball of $\ell_q^n$ for $q \geq 2$, when the dimension $n$ is sufficientlylarge, 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 resultfor $p=c \in (0,1)$, a universal constant. It turns out that the local versionof these conjectures is equivalent to a minimization problem for a spectral-gapparameter associated with a certain differential operator, introduced byHilbert (under different normalization) in his proof of the Brunn-Minkowskiinequality. As applications, we obtain local uniqueness results in the even$L^p$-Minkowski problem, as well as improved stability estimates in theBrunn-Minkowski and anisotropic isoperimetric inequalities.

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