Uncentered Blaschke-Santaló inequalities for the Gaussian measure

We study the maximizers of the generalized volume product $${\gamma }_{\sigma }^n(A){\gamma }_{\sigma }^n(A^{\circ })$$ among all measurable subsets $A\subset {\mathbb{R}}^n$, where $A^{\circ }$ denotes the polar set of $A$, and where ${\gamma }_{\sigma }^n$ denotes the centered Gaussian probability measure on ${\mathbb{R}}^n$ with covariance ${\sigma }^2{I}_n$, $\sigma >0$. It turns out that the maximizers depend on $\sigma$. 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 ${\sigma }^2\le \frac{1}{n}$ we show that the Euclidean unit ball is the unique maximizer, while this is no longer the case for ${\sigma }^2\ge \frac{2}{n+1}$. In dimension $n=2$, we close the gap by showing that the Euclidean unit ball is the unique maximizer for ${\sigma }^2\le \frac{2}{3}$.