实射影结构的凸性定理
A convexity theorem for real projective structures
摘要：
给定RP^n（n>1）中凸n-多面体的有限集合P，我们考虑一个实射影流形M，它是通过将P中的多面体沿其刻面粘合在一起而得到的，使得共享一个公共刻面的任何两个相邻多面体的并集是凸的。本文证明了M上的实射影结构是凸的，如果P不含三角多面体;是真凸的，如果P还含一个其对偶多面体是厚的多面体。三角形多面体和厚边多面体分别被定义为至少有五条边的三角形和多边形的类似物。
英文摘要：
Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on M is (1) convex if P contains no triangular polytope, and (2) properly convex if, in addition, P contains a polytope whose dual polytope is thick. Triangular polytopes and polytopes with thick duals are defined as analogues of triangles and polygons with at least five edges, respectively.
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