Counterexamples and symmetry for uneven orthogonal mass partitions in the plane
Abstract
Grünbaum asked whether every planar convex body admits, for every 0≤ t≤ 1/4, two orthogonal lines cutting it into pieces with cyclically ordered areas $t,t,1/2-t,1/2-t$. Bárány posed the analogous question for well-behaved planar measures and conjectured that the answer there is negative. We confirm Bárány's conjecture in a particularly robust form: for every fixed 0<t<1/4 we construct smooth, strictly positive, centrally symmetric, strongly log-concave measures arbitrarily close to the standard Gaussian for which the prescribed partition does not exist. In contrast, we prove that the partition exists for every t whenever the measure is invariant under an orientation-reversing affine involution. We also exhibit a $96$-point counterexample for which no pair of perpendicular lines produces cyclic counts $8,8,40,40$.
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BibTeX
@misc{martinezsandoval2026counterexamples,
title = {{Counterexamples and symmetry for uneven orthogonal mass partitions in the plane}},
author = {Leonardo Martínez-Sandoval},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.16757},
eprint = {2609.16757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16757},
}
RIS
TY - UNPB
TI - Counterexamples and symmetry for uneven orthogonal mass partitions in the plane
AU - Leonardo Martínez-Sandoval
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.16757
UR - https://arxiv.org/abs/2609.16757
AB - Grünbaum asked whether every planar convex body admits, for every 0≤ t≤ 1/4, two orthogonal lines cutting it into pieces with cyclically ordered areas $t,t,1/2-t,1/2-t$. Bárány posed the analogous question for well-behaved planar measures and conjectured that the answer there is negative. We confirm Bárány's conjecture in a particularly robust form: for every fixed 0<t<1/4 we construct smooth, strictly positive, centrally symmetric, strongly log-concave measures arbitrarily close to the standard Gaussian for which the prescribed partition does not exist. In contrast, we prove that the partition exists for every t whenever the measure is invariant under an orientation-reversing affine involution. We also exhibit a $96$-point counterexample for which no pair of perpendicular lines produces cyclic counts $8,8,40,40$.
ER -