A theorem of bárány revisited and extended
Nabil H. Mustafa, Saurabh Ray · 2012
The colorful Carathéodory theorem [B82] 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 D2+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 D2-dimensional rainbow simplex intersecting C.