Zong's problem on $2d$ half-spaces containing the unit ball

We prove that the intersection of $2d$ halfspaces containing the Euclidean unit ball in ${\mathbb{R}}^d$ contains a point of norm at least $\sqrt{d}$. Equality characterizes the cube. The proof gives a lower bound for the average squared norm of the vertices, weighted by the Gaussian measures of their normal cones. The main step is an inequality for Gaussian orthant probabilities, proved by a one-dimensional comparison of mass, centroid, and boundary density. We also discuss a quadratic extension and give an independent spectral estimate.