Counterexamples and symmetry for uneven orthogonal mass partitions in the plane

Leonardo Martínez-Sandoval

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arXiv · 2609.16757

Counterexamples and symmetry for uneven orthogonal mass partitions in the plane

Leonardo Martínez-Sandoval

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Abstract

Grünbaum asked whether every planar convex body admits, for every $0\leq t\leq 1/4$, 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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Leonardo Martínez-Sandoval. 2026-09-15. Counterexamples and symmetry for uneven orthogonal mass partitions in the plane. https://arxiv.org/abs/2609.16757

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