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Symmetry 2013, 5(1), 47-53; doi:10.3390/sym5010047
Short Note

A Note on Lower Bounds for Colourful Simplicial Depth

1,* , 2
 and
1
1 Advanced Optimization Laboratory, Department of Computing and Software, McMaster University, Hamilton, Ontario L8S 4K1, Canada 2 Department of Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
* Author to whom correspondence should be addressed.
Received: 18 October 2012 / Revised: 18 December 2012 / Accepted: 31 December 2012 / Published: 7 January 2013
(This article belongs to the Special Issue Polyhedra)
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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 colourful simplicial depth; Colourful Carathéodory Theorem; discrete geometry; polyhedra; combinatorial symmetry
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
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Deza, A.; Stephen, T.; Xie, F. A Note on Lower Bounds for Colourful Simplicial Depth. Symmetry 2013, 5, 47-53.

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