We show that for any even log-concave probability measure
μ
\mu
on
R
n
\mathbb {R}^n
, any pair of symmetric convex sets
K
K
and
L
L
, and any
λ
∈
[
0
,
1
]
\lambda \in [0,1]
,
μ
(
(
1
−
λ
)
K
+
λ
L
)
c
n
≥
(
1
−
λ
)
μ
(
K
)
c
n
+
λ
μ
(
L
)
c
n
,
\begin{equation*} \mu ((1-\lambda ) K+\lambda L)^{c_n}\geq (1-\lambda ) \mu (K)^{c_n}+\lambda \mu (L)^{c_n}, \end{equation*}
where
c
n
≥
n
−
4
−
o
(
1
)
c_n\geq n^{-4-o(1)}
. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures.