- Email: [email protected]
On Simplices Embracing a Point
Electronic Notes in Discrete Mathematics 31 (2008) 97 www.elsevier.com/locate/endm
On Simplices Embracing a Point J´anos Pach 1 City College, New York and R´enyi Institute,Budapest
A number of different problems in combinatorial and computational geometry lead to questions about systems of triangles in the plane or in space. However, we cannot answer even the simplest questions in geometric hypergraph theory. After surveying some outstanding open problems in this area, we prove the following theorem, established jointly with Holmsen and Tverberg: Let n > d + 1, let and let P = {p1 , . . . , pn } be a set of points in Rd whose convex hull contains the origin O in its interior. If P ∪ O is in general position, then there exists a d-tuple Q = {pi1 , . . . , pid } ⊂ P such that O is not contained in the convex hull of Q ∪ {p} for any p ∈ P \ Q. This answers a question of Strausz. In the process, we strengthen the Colored Helly and Caratheodory theorems, due to Lov´asz and B´ar´any, respectively. We also report on related recent results obtained in collaboration with Prasant Pal and Shannigrahi.
1
Email:[email protected]
1571-0653/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2008.06.019
On Simplices Embracing a Point J´anos Pach 1 City College, New York and R´enyi Institute,Budapest
A number of different problems in combinatorial and computational geometry lead to questions about systems of triangles in the plane or in space. However, we cannot answer even the simplest questions in geometric hypergraph theory. After surveying some outstanding open problems in this area, we prove the following theorem, established jointly with Holmsen and Tverberg: Let n > d + 1, let and let P = {p1 , . . . , pn } be a set of points in Rd whose convex hull contains the origin O in its interior. If P ∪ O is in general position, then there exists a d-tuple Q = {pi1 , . . . , pid } ⊂ P such that O is not contained in the convex hull of Q ∪ {p} for any p ∈ P \ Q. This answers a question of Strausz. In the process, we strengthen the Colored Helly and Caratheodory theorems, due to Lov´asz and B´ar´any, respectively. We also report on related recent results obtained in collaboration with Prasant Pal and Shannigrahi.
1
Email:[email protected]
1571-0653/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2008.06.019