点与平面的刺单形
Stabbing simplices by points and flats
摘要：
Barany在1982年证明了如下结果：对任意d >= 1，存在c_d > 0，使得对R^d中的任意n点集S，R ^d中的点p至少包含在S所张成的单形的c_dn ^{d+1} - O（n^d）中。我们研究了c_d的最大可能值。已知c_d <= 1/（2^d（d+1）！）(this估计实际上对每个点集S）都成立。我们构造了一个集合，证明了c_d <=（d+1）^{-（d+1）}，并且我们猜想这个估计是紧的。最著名的下界，由于瓦格纳，是c_d >= gamma_d：=（d^2+1）/（（d+1）！（d+1）^{d+1}）;在他的方法中，p可以被选为S的任何中心点。我们构造了一个中心点包含在不超过gamma_dn ^{d+1}+O（n^d）个由S生成的单形中的n-点集，从而表明使用任意中心点的方法不能进一步改进.我们还证明了对R^d中的每个n点集S，存在一个（d-2）-平坦，它至少刺到S所张成的三角形中的c_{d,d-2} n^3 - O（n^2）个，其中c_{d,d-2} >=（1/24）（1- 1/（2d-1）^2）.为此，我们建立了一个独立的均分结果（推广了Buck和Buck以及Ceder的平面结果）：R^d中的每个质量分布都可以被相交于一个公共（d-2）平面的2d-1个超平面分成4d-2等份。
英文摘要：
The following result was proved by Barany in 1982: For every d >= 1 there exists c_d > 0 such that for every n-point set S in R^d there is a point p in R^d contained in at least c_d n^{d+1} - O(n^d) of the simplices spanned by S. We investigate the largest possible value of c_d. It was known that c_d <= 1/(2^d(d+1)!) (this estimate actually holds for every point set S). We construct sets showing that c_d <= (d+1)^{-(d+1)}, and we conjecture this estimate to be tight. The best known lower bound, due to Wagner, is c_d >= gamma_d := (d^2+1)/((d+1)!(d+1)^{d+1}); in his method, p can be chosen as any centerpoint of S. We construct n-point sets with a centerpoint that is contained in no more than gamma_d n^{d+1}+O(n^d) simplices spanned by S, thus showing that the approach using an arbitrary centerpoint cannot be further improved. We also prove that for every n-point set S in R^d there exists a (d-2)-flat that stabs at least c_{d,d-2} n^3 - O(n^2) of the triangles spanned by S, with c_{d,d-2}>=(1/24)(1- 1/(2d-1)^2). To this end, we establish an equipartition result of independent interest (generalizing planar results of Buck and Buck and of Ceder): Every mass distribution in R^d can be divided into 4d-2 equal parts by 2d-1 hyperplanes intersecting in a common (d-2)-flat.
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