Uncentered Blaschke-Santaló inequalities for the Gaussian measure

RESUMO

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 $\sigma^2 \le \frac{2}{3}$.