An optimal generalization of the Colorful Carathéodory theorem

The Colorful Carathéodory theorem by Bárány (1982) states that given d+1 sets of points in R, 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 bd/2c + 1 sets of points in R 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 bd/2c-dimensional rainbow simplex intersecting C.