Uncentered Blaschke-Santaló inequalities for the Gaussian measure

Abstract: We study the maximizers of the generalized volume product [ γσ^n(A)\,γσ^n(A^\circ) ] among all measurable subsets $A\subset\mathbb{R}<sup>n$, where $A<sup>\circ$ denotes the polar set of $A$, and where $γ_σ<sup>n$ denotes the centered Gaussian probability measure on $\mathbb{R}<sup>n$ with covariance $σ² I_n$, $σ&gt;0$. It turns out that the maximizers depend on $σ$. We prove that they exist and are convex bodies. In dimension $n=1$, we find the exact form of the maximizers. In dimension $n\ge 2$, we show that they are smooth bodies of revolution whose support function satisfies a certain differential equation. Moreover, for $σ² \le \frac{1}{n}$ we show that the Euclidean unit ball is the unique maximizer, while this is no longer the case for $σ^2\ge {\frac{2}{n+1}}$. In dimension $n=2$, we close the gap by showing that the Euclidean unit ball is the unique maximizer for $σ² \le \frac{2}{3}$.