Colourful simplicial depth

Abstract

Inspired by Bárány's Colourful Carathéodory Theorem [4]. we introduce a colourful generalization of Liu's simplicial depth [13]. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d2 + 1 and that the maximum is dd+1 + 1. We exhibit configurations attaining each of these depths, and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.