  As one of the results of this manuscript, we show that for any log-concave
measure $\mu$ on $\mathbb{R}^n$, any pair of symmetric convex sets $K$ and $L$,
and any $\lambda\in [0,1],$ $$\mu((1-\lambda) K+\lambda L)^{c_n}\geq
(1-\lambda) \mu(K)^{c_n}+\lambda\mu(L)^{c_n},$$ where $c_n\geq n^{-4-o(1)}.$
  However, most of the paper is dedicated to studying sharp estimates in the
case of the standard Gaussian measure $\gamma$. We formulate a plausible
conjecture for the optimal Ehrhard-type inequality for convex symmetric sets
with respect to the Gaussian measure. Namely, letting $J_{k-1}(s)=\int^s_0
t^{k-1} e^{-\frac{t^2}{2}}dt$ and $c_{k-1}=J_{k-1}(+\infty)$, we conjecture
that the function $F:[0,1]\rightarrow\mathbb{R},$ given by $$F(a)= \sum_{k=1}^n
1_{a\in E_k}\cdot(\beta_k J_{k-1}^{-1}(c_{k-1} a)+\alpha_k)$$ (with some choice
of a decomposition $[0,1]=\cup_{i} E_i$ and coefficients $\alpha_i, \beta_i$)
satisfies, for all symmetric convex sets $K$ and $L,$ and any
$\lambda\in[0,1]$, $$ F\left(\gamma(\lambda K+(1-\lambda)L)\right)\geq \lambda
F\left(\gamma(K)\right)+(1-\lambda) F\left(\gamma(L)\right). $$ We explain that
this conjecture is "the most optimistic possible", and make progress towards
it. See more details in the abstract in the pdf.
  As an auxiliary result on the way to the equality case characterization, we
characterize the equality cases in the "convex set version" of the
Brascamp-Lieb inequality, and moreover, obtain a quantitative stability version
in the case of the standard Gaussian measure; this may be of independent
interest. All the equality case characterizations rely on the careful analysis
of the smooth case, the stability versions via trace theory, and local
approximation arguments.
  In addition, we provide a non-sharp estimate for a function $F$ whose
composition with $\gamma(K)$ is concave in the Minkowski sense for all
symmetric convex sets.
