A theorem of bárány revisited and extended

Nabil H. Mustafa, Saurabh Ray
2012 Proceedings of the 2012 symposuim on Computational Geometry - SoCG '12  
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 colorful
more » ... y theorem: given ⌊d/2⌋ + 1 sets of points in R d , 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.
doi:10.1145/2261250.2261300 dblp:conf/compgeom/MustafaR12 fatcat:6ttq7n37yrb33inhnc5bhfditq