A Further Generalization of the Colourful Carathéodory Theorem Frédéric Meunier and Antoine Deza

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 sufficient condition and obtain new colourful Carathéodory theorems. We also give an algorithm to find a colourful simplex containing 0 under the generalized condition. In the plane an alternate, and more general, proof using graphs is given. In addition, we observe that 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 inRd 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] providing a more general sufficient condition for the existence of a colourful simplex containing the origin 0. 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). We further generalize the sufficient condition for the existence of a colourful simplex containing the origin. Moreover, the proof, given in Section 2.1, provides an alternate and geometric proof for Theorem 1.2. Let −−→ xk 0 denote the ray originating from xk towards 0. Date: July 14, 2011. 2000 Mathematics Subject Classification. 52C45, 52A35.