An optimal generalization of the Colorful Carathéodory theorem

The Colorful Carathéodory theorem by Bárány (1982) states that given d+1d+1 sets of points in RdRd, 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+1d+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⌊d/2⌋+1 sets of points in RdRd and a convex object CC, then either one set can be separated from CC by a constant   (depending only on dd) number of hyperplanes, or there is a ⌊d/2⌋⌊d/2⌋-dimensional rainbow simplex intersecting CC.