A combinatorial approach to colourful simplicial depth

Abstract

ABSTRACT. The colourful simplicial depth conjecture states that any point in the convex hull of each of d +1 sets, or colours, of d +1 points in general position in R d is contained in at least d 2 +1 simplices with one vertex from each set. We verify the conjecture in dimension 4 and strengthen the known lower bounds in higher dimensions. These results are obtained using a combinatorial generalization of colourful point configurations called octahedral systems. We present properties of octahedral systems generalizing earlier results on colourful point configurations and exhibit an octahedral system which cannot arise from a colourful point configuration. The number of octahedral systems is also given.1. INTRODUCTION 1.1. Preliminaries. An n-uniform hypergraph is said to be n-partite if its vertex set is the disjoint union of n sets V 1 , . . . ,V n and each edge intersects each V i at exactly one vertex. Such an hypergraph is an (n + 1)-tuple (V 1 , . . . ,V n , E ) where E is the set of edges. An octahedral system Ω is an n-partite hypergraph (V 1 , . . . ,V n , E ) with |V i | ≥ 2 for i = 1, . . . , n and satisfying the following parity condition: the number of edges ofi =1 S i of points such that |X ∩ S i | = 2 for i = 1, . . . , d + 1, there is an even number of colourful simplices generated by X and containing the origin 0. Therefore, the hypergraph Ω = (V 1 , . . . ,V d +1 , E ), with V i = S i for i = 1, . . . , d + 1 and where the edges in E correspond to the colourful simplices containing 0 forms an octahedral system. This property motivated Bárány to suggest octahedral systems as a combinatorial generalization of colourful point configurations, see [8].Let µ(d ) denote the minimum number of colourful simplices containing 0 over all colourful point configurations satisfying 0 ∈ d +1 i =1 conv(S i ) and Stephen, and Xie [8] . The lower bound was slightly improved in dimension 4 to µ(4) ≥ 14 via a computational approach presented in [9].An octahedral system arising from a colourful point configuration S 1 , . . . ,and |S i | = d + 1 for all i , is without isolated vertex; that is, each vertex belongs to at least one edge. Indeed, according to a strengthening of the colourful Carathéodory theorem [2], any point of such a colourful configuration is the vertex of at least one colourful simplex containing 0. Theorem 1.1, whose proof is given in Section 4, provides a lower bound for the number of edges of an octahedral system without isolated vertex.

The colourful simplicial depth conjecture

“…The following proposition, proved in [6], states that the set of all octahedral systems is stable under the "symmetric difference" operation.…”

“…It was implicitly proved in Section 3 of [6] that any octahedral system can be described as a symmetric difference of umbrellas. In this paper, we describe an octahedral system as a symmetric difference of other octahedral systems to bound its cardinality.…”

“…Deza et. al [6] improved the bound to 1 2 d 2 + 7 2 d − 8 for d ≥ 4. This latter result was obtained using a combinatorial generalization of the colourful point configurations suggested by Bárány and known as octahedral systems, see [5].…”

A Note on Lower Bounds for Colourful Simplicial Depth

“…This improves by 1 the bound of Deza et al [5], from µ(4) ≥ 13 to µ(4) ≥ 14. Since this article was written, Deza et al [17] have introduced a different approach that shows µ(4) = 17 and improves the bounds in higher dimension as well.…”

Colorful Simplicial Depth, Minkowski Sums, and Generalized Gale Transforms

“…The initial lower bound of 2d from [7] was improved in a series of papers [4,8,9,26] culminating in the resolution of the conjectured lower bound by Sarrabezolles [24]. In [7] also a conjectured upper bound was proposed.…”