118 research outputs found

    Colourful Simplicial Depth

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    Inspired by Barany's colourful Caratheodory theorem, we introduce a colourful generalization of Liu's simplicial depth. 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.Comment: 18 pages, 5 figues. Minor polishin

    The colourful simplicial depth conjecture

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    Given d+1d+1 sets of points, or colours, S1,…,Sd+1S_1,\ldots,S_{d+1} in Rd\mathbb R^d, a colourful simplex is a set T⊆⋃i=1d+1SiT\subseteq\bigcup_{i=1}^{d+1}S_i such that ∣T∩Si∣≤1|T\cap S_i|\leq 1, for all i∈{1,…,d+1}i\in\{1,\ldots,d+1\}. The colourful Carath\'eodory theorem states that, if 0\mathbf 0 is in the convex hull of each SiS_i, then there exists a colourful simplex TT containing 0\mathbf 0 in its convex hull. Deza, Huang, Stephen, and Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597--604 (2006)) conjectured that, when ∣Si∣=d+1|S_i|=d+1 for all i∈{1,…,d+1}i\in\{1,\ldots,d+1\}, there are always at least d2+1d^2+1 colourful simplices containing 0\mathbf 0 in their convex hulls. We prove this conjecture via a combinatorial approach

    Algorithms for Colourful Simplicial Depth and Medians in the Plane

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    The colourful simplicial depth of a point x in the plane relative to a configuration of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point, called a median, that maximizes colourful simplicial depth. For computing the colourful simplicial depth of x, our algorithm runs in time O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For finding the colourful median, we get a time of O(n^4). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O(n log(n)) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n^4) for finding a monochrome median.Comment: 17 pages, 8 figure

    A Tverberg type theorem for matroids

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    Let b(M) denote the maximal number of disjoint bases in a matroid M. It is shown that if M is a matroid of rank d+1, then for any continuous map f from the matroidal complex M into the d-dimensional Euclidean space there exist t \geq \sqrt{b(M)}/4 disjoint independent sets \sigma_1,\ldots,\sigma_t \in M such that \bigcap_{i=1}^t f(\sigma_i) \neq \emptyset.Comment: This article is due to be published in the collection of papers "A Journey through Discrete Mathematics. A Tribute to Jiri Matousek" edited by Martin Loebl, Jaroslav Nesetril and Robin Thomas, due to be published by Springe

    A Note on Lower Bounds for Colourful Simplicial Depth

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    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
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