Counterexamples and symmetry for uneven orthogonal mass partitions in the plane

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 $8,8,40,40$.