关于局部凸PL流形与凸性的快速验证
度量几何
2007-05-23 v2
摘要
我们证明了：n维欧氏空间（n>2）中一个闭连通(n-1)维PL流形的实现是某个凸多面体的边界，当且仅当每个(n-3)维面的内部存在一点，该点有一个邻域落在某个凸n维体的边界上。这一结果源于将Van Heijenoort关于局部凸流形的定理推广到球面空间。我们还简要分析了在非紧致双曲空间中，局部凸性、曲面拓扑与整体凸性之间的关系。我们的PL流形凸性判据意味着，对于给定维数n>2的欧氏或球面空间中的闭紧致PL曲面，存在一种简单的多项式时间算法来检验其凸性。
引用
@article{arxiv.math/0309370,
title = {On locally convex PL-manifolds and fast verification of convexity},
author = {Konstantin Rybnikov},
journal= {arXiv preprint arXiv:math/0309370},
year = {2007}
}
备注
10 pages (abbreviated version). Significantly different from all older versions. Discount the previous version -- it had many omissions and typos, like the following one: indeed, everything works starting from dimension n=3, not n=2 as was printed in the old abstract. Hyperbolic and spherical cases have been substantially rewritten and errors fixed. This preprint is close to a similar preprint on the CS part of arxiv.org