Points surrounding the origin

Suppose d > 2, n > d+1, and we have a set P of n points in d-dimensional Euclidean space. Then P contains a subset Q of d points such that for any p ∈ P, the convex hull of Q∪{p} does not contain the origin in its interior. We also show that for non-empty, finite point sets A1, ..., Ad+1 in ℝd, if the origin is contained in the convex hull of Ai ∪ Aj for all 1≤i<j≤d+1, then there is a simplex S containing the origin such that |S∩Ai|=1 for every 1≤i≤d+1. This is a generalization of Bárány’s colored Carathéodory theorem, and in a dual version, it gives a spherical version of Lovász’ colored Helly theorem.


Published in:
Combinatorica, 28, 6, 633–644
Year:
2008
Note:
Professor Pach's number: [216]
Laboratories:




 Record created 2008-11-18, last modified 2018-09-13

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