论文信息 - A Tverberg-type result on multicolored simplices

A Tverberg-type result on multicolored simplices

Abstract Let P 1 , P 2 ,…, P d +1 be pairwise disjoint n - element point sets in general position in d - space . It is shown that there exist a point O and suitable subsets Q i ⊆ P i ( i = 1, 2,…, d + 1) such that  Q i  ≥ c d  P i , an every d - dimensional simplex with exactly one vertex in each Q i contains Q in its interior. Here c d is a positive constant depending only on d .

János Pach

[1] Karol Borsuk. Drei Sätze über die n-dimensionale euklidische Sphäre , 1933 .

[2] I. Bárány,et al. A Colored Version of Tverberg's Theorem , 1992 .

[3] R. Pollack,et al. Geometric Transversal Theory , 1993 .

[4] H. Tverberg. A Generalization of Radon's Theorem , 1966 .

[5] J. Radon. Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten , 1921 .

[6] E. Szemerédi. Regular Partitions of Graphs , 1975 .

[7] Rephael Wenger,et al. Bounding the number of geometric permutations induced by k-transversals , 1994, SCG '94.

[8] Noga Alon,et al. Point Selections and Weak ε-Nets for Convex Hulls , 1992, Combinatorics, Probability and Computing.

[9] Gil Kalai,et al. Intersection patterns of convex sets , 1984 .

[10] Vojtech Rödl,et al. The Uniformity Lemma for hypergraphs , 1992, Graphs Comb..

[11] Fan Chung Graham,et al. Regularity Lemmas for Hypergraphs and Quasi-randomness , 1991, Random Struct. Algorithms.

[12] Zoltán Füredi,et al. On the number of halving planes , 1989, SCG '89.

[13] Sinisa T. Vrecica,et al. The Colored Tverberg's Problem and Complexes of Injective Functions , 1992, J. Comb. Theory, Ser. A.

[14] Rephael Wenger,et al. Bounding the Number of Geometric Permutations Induced byk-Transversals , 1996, J. Comb. Theory, Ser. A.

[15] M. Katchalski,et al. A problem of geometry in ⁿ , 1979 .