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.