The colourful simplicial depth conjecture

Abstract

Given d + 1 sets of points, or colours, S 1 , . . . ,The colourful Carathéodory theorem states that, if 0 is in the convex hull of each S i , then there exists a colourful simplex T containing 0 in its convex hull. Deza, Huang, Stephen, and Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597-604 (2006)) conjectured that, when |S i | = d + 1 for all i ∈ {1, . . . , d + 1}, there are always at least d 2 + 1 colourful simplices containing 0 in their convex hulls. We prove this conjecture via a combinatorial approach.

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

“…Define the colorful simplicial depth of P , denoted ColorfulSimp-Depth(P ), as the number of colorful simplices in P containing o. Deza et al [138] proposed some lower bounds on the colorful simplicial depth, and they conjectured that if |P i | = d + 1 for 0 ≤ i ≤ d, then ColorfulSimp-Depth(P ) ≥ d 2 + 1. This was proven by Sarrabezolles [335]. The bound is optimal by the work of Deza et al [138].…”

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.…”

Algorithms for Colourful Simplicial Depth and Medians in the Plane

“…Deza et al [DHST06] formalized the notion and considered bounds for the colourful depth of points in the intersection of the convex hulls of the colours. Among the recent work on colourful depth are proofs of the lower [Sar15] The monochrome simplicial depth can be computed by enumerating simplices, but in general dimension, it is quite challenging to compute it more efficiently [Alo06], [CO01], [FR05]. Several authors have considered the two-dimensional version of the problem, including Khuller and Mitchell [KM90], Gil, Steiger and Wigderson [GSW92] and Rousseeuw and Ruts [RR96].…”

“…Deza et al [DHST06] formalized the notion and considered bounds for the colourful depth of points in the intersection of the convex hulls of the colours. Among the recent work on colourful depth are proofs of the lower [Sar15] and upper [ABP + 16] bounds conjectured by Deza et al, with the latter result showing beautiful connections to Minkowski sums of polytopes.…”