118 research outputs found
Colourful Simplicial Depth
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
Given sets of points, or colours, in , a colourful simplex is a set such that
, for all . The colourful
Carath\'eodory theorem states that, if is in the convex hull of
each , then there exists a colourful simplex containing in
its convex hull. Deza, Huang, Stephen, and Terlaky (Colourful simplicial depth,
Discrete Comput. Geom., 35, 597--604 (2006)) conjectured that, when
for all , there are always at least colourful
simplices containing in their convex hulls. We prove this
conjecture via a combinatorial approach
Algorithms for Colourful Simplicial Depth and Medians in the Plane
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
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
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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