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Zong's problem on 2d half-spaces containing the unit ball
Abstract
We prove that the intersection of 2d halfspaces containing the Euclidean unit ball in ℝd contains a point of norm at least √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.
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BibTeX
@misc{ivanov2026zongs,
title = {{Zong's problem on $2d$ half-spaces containing the unit ball}},
author = {Grigory Ivanov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38394},
eprint = {2609.38394},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38394},
}
RIS
TY - UNPB
TI - Zong's problem on 2d half-spaces containing the unit ball
AU - Grigory Ivanov
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38394
UR - https://arxiv.org/abs/2609.38394
AB - We prove that the intersection of 2d halfspaces containing the Euclidean unit ball in ℝ^d contains a point of norm at least √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.
ER -