A quadratic lower bound for colourful simplicial depth

Stephen, Tamon; Thomas, Hugh

doi:10.1007/s10878-008-9149-x

Cited by 13 publications

(13 citation statements)

References 6 publications

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“…The lower bound for µ(d) was improved very recently independently by Bárány and Matoušek [5] and Stephen and Thomas [15] to max(3d, 1 5 d(d + 1)) for d > 2 and (d + 2) 2 /4 respectively. We know that µ(1) = 2, µ(2) = 5 and µ (3) We focus on the case where we have (d + 1) colours.…”

Section: Discussionmentioning

confidence: 97%

Colourful Simplicial Depth

Deza

Huang

Stephen

et al. 2006

Discrete Comput Geom

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Inspired by Bárány's Colourful Carathéodory Theorem [4], we introduce a colourful generalization of Liu's simplicial depth [13]. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d 2 + 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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Section: Discussionmentioning

confidence: 97%

Colourful Simplicial Depth

Deza

Huang

Stephen

et al. 2006

Discrete Comput Geom

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“…They proved that 2d ≤ µ(d) ≤ d 2 + 1 and conjectured that µ(d) = d 2 + 1. Later I. Bárány and J. Matoušek [3] proved that µ(d) ≥ max 3d, d(d+1) 5 for d ≥ 3, Stephen and Thomas [4] proved that…”

Section: Introductionmentioning

confidence: 99%

The colourful simplicial depth conjecture

Sarrabezolles

2015

Journal of Combinatorial Theory, Series A

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Given d + 1 sets of points, or colours, S 1 , . . . ,The colourful Carathéodory theorem states that, if 0 is in the convex hull of each S i , then there exists a colourful simplex T containing 0 in its convex hull. Deza, Huang, Stephen, and Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597-604 (2006)) conjectured that, when |S i | = d + 1 for all i ∈ {1, . . . , d + 1}, there are always at least d 2 + 1 colourful simplices containing 0 in their convex hulls. We prove this conjecture via a combinatorial approach.

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“…when d is odd. The lower bound has since been improved by Bárány and Matoušek [3] (who verified the conjecture for d = 3), Stephen and Thomas [4] and Deza et al [5], which includes the current strongest…”

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confidence: 95%

A Note on Lower Bounds for Colourful Simplicial Depth

2013

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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 d 2 + 1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound⌉. 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 Classification: MSC 52A35, 05D05, 52B05A colourful configuration is the union of (d+1) sets, or colours,Without loss of generality we assume that the points in S ∪ {0} are in general position. We are interested in the colourful simplices formed by taking the convex hull of a set containing one point of each colour. The colourful simplicial depth problem is to find a colourful configuration, with each S i containing the origin 0 in the interior of its convex hull, minimizing the number of colourful simplices containing 0. We denote this minimum by µ(d). We take the simplices to be closed and remark that the minimum should be attained.

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A quadratic lower bound for colourful simplicial depth

Abstract: Abstract. We show that any point in the convex hull of each of (d + 1) sets of (d + 1) points in R d is contained in at least (d + 2) 2 /4 simplices with one vertex from each set.

Cited by 13 publications

References 6 publications

Colourful Simplicial Depth

Colourful Simplicial Depth

The colourful simplicial depth conjecture

A Note on Lower Bounds for Colourful Simplicial Depth

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