Uncentered Blaschke-Santaló inequalities for the Gaussian measure

We study the maximizers of the generalized volume product \[ γ_σ^n(A)\,γ_σ^n(A^\circ) \] among all measurable subsets $A\subset\mathbb{R}^n$, where $A^\circ$ denotes the polar set of $A$, and where $γ_σ^n$ denotes the centered Gaussian probability measure on $\mathbb{R}^n$ with covariance $σ^2 I_n$, $σ>0$. 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 $σ^2 \le \frac{1}{n}$ 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 $σ^2 \le \frac{2}{3}$.