Any vector is assumed to be a column vector by default. Suppose $f(\mathbf{x})$ is the $d$-dimensional standard Gaussian density. I am interested in the following optimization problem: $$ \max_g ~\det \left(\int_{\mathbb{R}^d} \mathbf{x}\mathbf{x}^T f(\mathbf{x}) g(\mathbf{x}) d \mathbf{x}\right), $$ where $\det$ stands for the determinant, and the maximization is taken over all measurable $g$ such that $0\le g\le 1$ and $$ \int_{\mathbb{R}^d} f(\mathbf{x})g(\mathbf{x})d\mathbf{x}=1/2. $$
Remark 1: My conjecture is that the maximizer $g$ is the indicator function of the complement of a ball in $\mathbb{R}^d$ of a suitable radius, but I could not find a proof.
Remark 2: It would be even useful to only prove that this optimization problem has a solution, namely, there exists such a $g$ which attains the maximum.