On a conjectural symmetric version of Ehrhard's inequality

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 $F(a)= \sum_{k=1}^n 1_{a\in E_k}\cdot(β_k J_{k-1}^{-1}(c_{k-1} a)+α_k)$ (with an appropriate choice of a decomposition $[0,1]=\cup_{i} E_i$ and coefficients $α_i, β_i$) satisfies, for all symmetric convex sets $K$ and $L,$ and any $λ\in[0,1]$, $F\left(γ(λK+(1-λ)L)\right)\geq λF\left(γ(K)\right)+(1-λ) F\left(γ(L)\right).$ We explain that this conjecture is "the most optimistic possible", and is equivalent to the fact that for any symmetric convex set $K,$ its \emph{Gaussian concavity power} $p^s(K,γ)$ is greater than or equal to $p_s(RB^k_2\times \mathbb{R}^{n-k},γ),$ for some $k\in \{1,...,n\}$. We call the sets $RB^k_2\times \mathbb{R}^{n-k}$ round $k$-cylinders; they also appear as the conjectured Gaussian isoperimetric minimizers for symmetric sets, see Heilman \cite{Heilman}. In this manuscript, we make progress towards this question, and prove certain inequality for which the round k-cylinders are the only equality cases. 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.

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