A further generalization of the colourful Carathéodory theorem

Given d +1 sets, or colours, S1,S2, . . . ,Sd+1 of points in Rd , a colourful set is a set S ⊂⋃i Si such that |S ∩Si | ≤ 1 for i = 1, . . . ,d +1. The convex hull of a colourful set S is called a colourful simplex. Bárány’s colourful Carathéodory theorem asserts that if the origin 0 is contained in the convex hull of Si for i = 1, . . . ,d + 1, then there exists a colourful simplex containing 0. The sufficient condition for the existence of a colourful simplex containing 0 was generalized to 0 being contained in the convex hull of Si ∪ S j for 1 ≤ i < j ≤ d + 1 by Arocha et al. and by Holmsen et al. We further generalize the theorem by showing that a colourful simplex containing 0 exists if, for 1 ≤ i < j ≤ d +1, there exists k ∉ {i , j } such that, for all xk ∈ Sk , the convex hull of Si ∪ S j intersects the ray −−→ xk 0 in a point distinct from xk . A slightly stronger version of this new colourful Carathéodory theorem is also given. This result provides a short and geometric proof of the previous generalization of the colourful Carathéodory theorem. We also give an algorithm to find a colourful simplex containing 0 under the generalized condition. In the plane an alternative and more general proof using graphs is given. In addition, we observe that, in general, the existence of one colourful simplex containing 0 implies the existence of at least mini |Si | colourful simplices containing 0. In other words, any condition implying the existence of a colourful simplex containing 0 actually implies the existence of mini |Si | such simplices. 1. COLOURFUL CARATHÉODORY THEOREMS Given d + 1 sets, or colours, S1,S2, . . . ,Sd+1 of points in Rd , we call a set of points drawn from the Si ’s colourful if it contains at most one point from each Si . A colourful simplex is the convex hull of a colourful set S, and a colourful set of d points which misses Si is called an î -transversal. The colourful Carathéodory Theorem 1.1 by Bárány provides a sufficient condition for the existence of a colourful simplex containing the origin 0. Theorem 1.1 ([Bár82]). Let S1,S2, . . . ,Sd+1 be finite sets of points in Rd such that 0 ∈ conv(Si ) for i = 1. . .d +1. Then there exists a set S ⊂ ⋃i Si such that |S ∩Si | = 1 for i = 1, . . . ,d +1 and 0 ∈ conv(S). Theorem 1.1 was generalized by Arocha et al. [ABB+09] and by Holmsen et al. [HPT08] who provided a more general sufficient condition for the existence of a colourful simplex containing the origin. Theorem 1.2 ([ABB+09, HPT08]). Let S1,S2, . . . ,Sd+1 be finite sets of points in Rd such that 0 ∈ conv(Si ∪S j ) for 1 ≤ i < j ≤ d +1. Then there exists a set S ⊂⋃i Si such that |S ∩Si | = 1 for i = 1, . . . ,d +1 and 0 ∈ conv(S). Date: July 14, 2011. 2000 Mathematics Subject Classification. 52C45, 52A35.