多胞体的组合切片问题：如何（不）从切片重构多胞体
Combinatorial slicing problems of polytopes: How (not) to reconstruct a polytope from its slices
AI总结：
本文研究多胞体超平面截面的组合信息能否决定原多胞体组合类型，发现一般不能，但对足够一般的简单多胞体，中心截面的顶点数函数可决定完整组合类型，并证明组合版Busemann-Petty和Bourgain问题在所有维度均失败。
AI中文摘要：
我们研究多胞体超平面截面的组合方面，重点关注截面的组合结构在多大程度上决定原多胞体的组合结构。我们证明，一般而言，关于截面的组合信息不足以确定多胞体的f-向量。相反，对于足够一般的简单多胞体，记录每个中心截面的顶点数的函数决定了完整的组合类型。我们将不同的组合切片性质组织成层级结构，分别针对仿射截面和中心截面，根据它们在多胞体上确定的组合结构水平进行分类。与经典的度量切片问题类比，我们通过用面数替换体积，提出了Busemann-Petty问题和Bourgain切片问题的组合类比，并证明它们在每个维度和每个面维度上都失败。
英文摘要：
We study combinatorial aspects of hyperplane sections of polytopes, focusing on how much the combinatorics of the sections determines the combinatorics of the original polytope. We show that, in general, combinatorial information about the sections is not enough to determine even the $f$-vector of the polytope. In contrast, for sufficiently generic simple polytopes, the function recording the number of vertices of each central section determines the full combinatorial type. We organize different combinatorial slicing properties into hierarchies, separately for affine and central sections, according to the level of combinatorial structure they determine on the polytope. In analogy with classical metric slicing problems, we formulate combinatorial analogues of the Busemann-Petty problem and Bourgain's slicing problem by replacing volume with face numbers, and show that they fail in every dimension and every face dimension.
