Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Ivanov, Grigory
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215068578