Points Surrounding the Origin

For d > 2 and n > d+1, let P = { p₁, . . . , p_n } be a set of points in R^d whose convex hull contains the origin O in its interior. We show that if P ∪ O is in general position, then there exists a d-tuple Q = { p_i1, . . . , p_id } ⊂ P such that O is not contained in the convex hull of Q ∪ {p} for any p ∈ P \ Q. Generalizations of this property are also considered.

We also show that for disjoint, non-empty, finite point sets A₁, . . . , A_d+1 in R^d in general position with respect to the origin, if the origin is contained in the convex hull of A_i ∪ A_j for all 1 ≤ i < j ≤ d+1, then there is a simplex S containing the origin such that |S ∩ A_i| = 1 for every 1 ≤ i ≤ d+1. This is a generalization of Bárány's colored Carathéodory theorem, and dually, it gives a spherical version of Lovász' colored Helly theorem.

Joint work with Andreas Holmsen and Helge Tverberg.

Event Topic

Discrete Applied Math Seminar