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

A Note on Lower Bounds for Colourful Simplicial Depth

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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)
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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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MDPI and ACS Style

Deza, A.; Stephen, T.; Xie, F. A Note on Lower Bounds for Colourful Simplicial Depth. Symmetry 2013, 5, 47-53.

AMA Style

Deza A, Stephen T, Xie F. A Note on Lower Bounds for Colourful Simplicial Depth. Symmetry. 2013; 5(1):47-53.

Chicago/Turabian Style

Deza, Antoine; Stephen, Tamon; Xie, Feng. 2013. "A Note on Lower Bounds for Colourful Simplicial Depth." Symmetry 5, no. 1: 47-53.

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