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The colorful Carathéodory theorem [Bár82] states that given d + 1 sets of points in R d , 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 colorfuldoi:10.1145/2261250.2261300 dblp:conf/compgeom/MustafaR12 fatcat:6ttq7n37yrb33inhnc5bhfditq