Quadratically Many Colorful Simplices

Abstract

The colorful Carathéodory theorem asserts that if X 1 , X 2 , . . . , X d+1 are sets in R d , each containing the origin 0 in its convex hull, then exists a set S ⊆ X 1 ∪ · · · ∪ X d+1 with |S ∩ X i | = 1 for all i = 1, 2, . . . , d + 1 and 0 ∈ conv(S) (we call conv(S) a colorful covering simplex). Deza, Huang, Stephen and Terlaky proved that if the X i are in general position with respect to 0 (consequently, each X i has at least d + 1 points), then there are at least 2d colorful covering simplices, and they constructed an example with no more than d 2 + 1 such simplices. Under the same assumption, we show that there are at least 1 5 d(d + 1) colorful covering simplices, thus determining the order of magnitude. We also obtain a lower bound of 3d, which is better for small d, and in particular, together with a parity argument it settles the case d = 3, where the minimum possible number of colorful covering simplices is 10.

A Further Generalization of the Colourful Carathéodory Theorem

“…Note that Proposition 3.1 can also be proven by a degree argument on the map embedding the join of the S * i in R d , or using the Octahedron Lemma [BM07]. 3.2.…”

“…Colourful simplices containing 0 are generated whenever the antipode of a point of colour i is covered by an i -transversal. In particular, one can consider combinatorial octahedra generated by pairs of disjoint i -transversals, and rely on the fact that every octahedron Ω either covers all of S d −1 with colourful cones, or every point x ∈ S d −1 that is covered by colourful cones from Ω is covered by at least two distinct such cones, see for example the Octahedron Lemma of [BM07]. One of the key argument in the proof of Theorem 1.4 can be reformulated as: either the pair of d + 1-transversals (T, T ) forms a octahedron covering S d −1 , or 0 belongs to a colourful simplex having conv(T ) as a facet.…”

A Note on Lower Bounds for Colourful Simplicial Depth

“…This is a topological fact that corresponds to the fact that 0 is either inside or outside the octahedron, see the Octahedron Lemma of [3] for a proof. Figure 1 illustrates this in a two-dimensional case where 0 is at the centre of a circle that contains points of the three colours.…”

“…when d is odd. The lower bound has since been improved by Bárány and Matoušek [3] (who verified the conjecture for d = 3), Stephen and Thomas [4] and Deza et al [5], which includes the current strongest…”

The colourful simplicial depth conjecture

“…They proved that 2d ≤ µ(d) ≤ d 2 + 1 and conjectured that µ(d) = d 2 + 1. Later I. Bárány and J. Matoušek [3] proved that µ(d) ≥ max 3d, d(d+1) 5 for d ≥ 3, Stephen and Thomas [4] proved that…”