On a conjectural symmetric version of Ehrhard's inequality

TL;DR

This paper proposes a conjecture for an optimal symmetric Ehrhard-type inequality under Gaussian measure, introduces a new function to describe it, and makes progress by proving related inequalities and characterizing equality cases.

Contribution

It formulates a conjecture for the symmetric Ehrhard inequality, introduces a novel function framework, and proves related inequalities with equality case characterizations.

Findings

Proved certain inequalities with round k-cylinders as equality cases.

Characterized equality cases in the convex set version of the Brascamp-Lieb inequality.

Provided a quantitative stability result for the Gaussian measure case.

Abstract

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 an appropriate 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 is equivalent to the fact that for any symmetric convex set $K,$ its \emph{Gaussian concavity power} $p^s(K,\gamma)$ is greater than or equal to $p_s(RB^k_2\times \mathbb{R}^{n-k},\gamma),$ 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.