平面中不均匀正交质量划分的反例与对称性
Counterexamples and symmetry for uneven orthogonal mass partitions in the plane
- Faculty of Sciences, UNAM（墨西哥国立自治大学理学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究证实了Bárány关于平面测度正交划分的猜想，构造了接近高斯的反例，并证明了在保向仿射对合下划分存在，同时给出96点反例。
AI中文摘要：
Grünbaum 提出如下问题：每个平面凸体是否对每个 $0\leq t\leq 1/4$ 都存在两条正交直线将其切割成四块，且按循环顺序的面积比为 $t,t,1/2-t,1/2-t$。Bárány 对性质良好的平面测度提出了类似问题，并猜想其答案是否定的。我们以特别稳健的形式证实了 Bárány 的猜想：对每个固定的 $0<t<1/4$，我们构造了任意接近标准高斯分布的平滑、严格正、中心对称、强对数凹测度，使得上述指定的划分不存在。相反，我们证明当测度在保向仿射对合下不变时，对每个 $t$ 该划分都存在。我们还构造了一个 96 点的反例，其中不存在一对垂直直线产生循环计数 $8,8,40,40$。
英文摘要：
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 $8,8,40,40$.
