2012/12/19 by Antoine Deza, Deza, Antoine, Frédéric Meunier +3
Computer Science · Engineering · #05C65 (Primary) 52C45 #52A35 (Secondary) #Combinatorics (math.CO) #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Optimization and Packing Problems #graph theory and CDMA systems
paper · doi:10.48550/arxiv.1212.4720
openalex publication_date 2012/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Rd is contained in at least d2+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 can not arise from a colourful point configuration. The number of octahedral systems is also given.