R~n中构形和单纯复形的一个最优度量不等式

对于R~n中一般位置的点构形,定义了第r个极小凸包距离的概念,证明了极小凸包距离和极小点-超平面距离之间的一个最优不等式.该不等式的一个直接推论是:对于R~n中一个k-维单纯复形K,我们能用其顶点集的极小点-超平面距离下估计K的Gromov-Guth厚度.进一步,在每一个维数k,构造了例子说明该下界几乎是最优的.

An Optimal Metric Inequality for Configurations and Simplicial Complexes in Rn

For a point configuration in general position in Rⁿ,we define the notion of the r-th minimal convex hull distance and prove an optimal inequality between the minimal convex hull distance and the minimal point-hyperplane distance.A direct corollary of this result is that we can estimate from below the Gromov-Guth thickness of a k-dimensional simplicial complex K in Rⁿ in terms of the minimal point-hyperplane distance of the vertex set of K.Moreover,we construct an example which shows that this lower bound is almost optimal in every dimension k.