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A Note on Lower Bounds for Colourful Simplicial Depth
Advanced Optimization Laboratory, Department of Computing and Software, McMaster University, Hamilton, Ontario L8S 4K1, Canada
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
* Author to whom correspondence should be addressed.
Received: 18 October 2012; in revised form: 18 December 2012 / Accepted: 31 December 2012 / Published: 7 January 2013
(This article belongs to the Special Issue Polyhedra
Abstract: The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d2 + 1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound for d ≥ 4 is ⌈(d+1)2 /2 ⌉. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14.
Keywords: colourful simplicial depth; Colourful Carathéodory Theorem; discrete geometry; polyhedra; combinatorial symmetry
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Deza, A.; Stephen, T.; Xie, F. A Note on Lower Bounds for Colourful Simplicial Depth. Symmetry 2013, 5, 47-53.
Deza A, Stephen T, Xie F. A Note on Lower Bounds for Colourful Simplicial Depth. Symmetry. 2013; 5(1):47-53.
Deza, Antoine; Stephen, Tamon; Xie, Feng. 2013. "A Note on Lower Bounds for Colourful Simplicial Depth." Symmetry 5, no. 1: 47-53.