Colourful Simplicial Depth

2005/06/01 by Antoine Deza, Deza, Antoine, Sui Huang +5
Engineering · Mathematics · #52A35 #52C45 #Advanced Optimization Algorithms Research #Advanced Statistical Methods and Models #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems #math.CO #msc:52A35 #msc:52C45

paper · pdf · doi:10.48550/arxiv.math/0506003

18 pages, 5 figues. Minor polishing

openalex publication_date 2005/06/01 · arxiv created 2006/01/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Inspired by Barany's colourful Caratheodory theorem, we introduce a colourful generalization of Liu's simplicial depth. 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 d^(d+1)+1. We exhibit configurations attaining each of these depths and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.

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