标题： 彩色单纯形深度，Minkowski和，以及广义Gale变换 显示英文标题

标题： Colorful simplicial depth, Minkowski sums, and generalized Gale transforms

摘要： 一种在欧几里得d维空间中d+1个有限点集的彩色单纯形深度是选择每个集合中的一个点使得原点包含在它们的凸包中的方式数目。我们使用组合拓扑的方法证明了彩色单纯形深度的紧上界。这证实了Deza等人(2006)的猜想。此外，我们引入了彩色Gale变换作为彩色配置和Minkowski和之间的桥梁。我们的彩色上界进而给出了某些单纯形Minkowski和的完全混合面数的紧上界。这解决了正常曲面理论中Burton(2003)的猜想。 显示英文摘要

摘要： The colorful simplicial depth of a collection of d+1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial topology to prove a tight upper bound on the colorful simplicial depth. This implies a conjecture of Deza et al. (2006). Furthermore, we introduce colorful Gale transforms as a bridge between colorful configurations and Minkowski sums. Our colorful upper bound then yields a tight upper bound on the number of totally mixed facets of certain Minkowski sums of simplices. This resolves a conjecture of Burton (2003) in the theory of normal surfaces.