  The $L^p$-Brunn-Minkowski theory for $p\geq 1$, proposed by Firey and
developed by Lutwak in the 90's, replaces the Minkowski addition of convex sets
by its $L^p$ counterpart, in which the support functions are added in
$L^p$-norm. Recently, B\"{o}r\"{o}czky, Lutwak, Yang and Zhang have proposed to
extend this theory further to encompass the range $p \in [0,1)$. 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 $\mathbb{R}^n$ and $p \in [1
- \frac{c}{n^{3/2}},1)$. In addition, we confirm the local log-Brunn--Minkowski
conjecture (the case $p=0$) for small-enough $C^2$-perturbations of the
unit-ball of $\ell_q^n$ for $q \geq 2$, 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 $\ell_q^n$ with $q \in [1,2)$, we confirm an analogous result
for $p=c \in (0,1)$, 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
$L^p$-Minkowski problem, as well as improved stability estimates in the
Brunn-Minkowski and anisotropic isoperimetric inequalities.

Functional Analysis;Spectral Theory