An optimal generalization of the Colorful Carathéodory theorem

Abstract

The Colorful Carathéodory theorem by Bárány (1982) states that given d+1 sets of points in Rd, the convex hull of each containing the origin, there exists a simplex (called a ‘rainbow simplex’) with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d+1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the Colorful Carathéodory theorem: given ⌊d/2⌋+1 sets of points in Rd and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a ⌊d/2⌋-dimensional rainbow simplex intersecting C.