Points surrounding the origin

Abstract

For d > 2 and n > d + 1, let P = {p 1 , . . . , p n } be a set of points in R d whose convex hull contains the origin O in its interior. We show that if P ∪ {O} is in general position, then there exists a d-tuple Q = {p i 1 , . . . , p i d } ⊂ P such that O is not contained in the convex hull of Q ∪ {p} for any p ∈ P \ Q. Generalizations of this property are also considered.We also show that for non-empty, finite point sets A 1 , . . . , A d+1 in R d , if the origin is contained in the convex hull of A i ∪A j for all 1 ≤ i < j ≤ d +1, then there is a simplex S containing the origin such that |S ∩ A i | = 1 for every 1 ≤ i ≤ d + 1. This is a generalization of Bárány's colored Carathéodory theorem, and dually, it gives a spherical version of Lovász' colored Helly theorem.

Matching points with disks with a common intersection

“…If C p 2p2 contains s 13 , then we are done, since s 13 is also common to D p 1p1 and D p 3p3 . Hence, assume C p 2p2 does not contain s 13 . Under this assumption, by Lemma 4 (used with points A = p 1 , B = p 2 , P = s 12 , and R = s 13 ), O and p 2 are in the same half-plane H bounded by (p 1 , p 3 ).…”

Carathéodory-type Theorems à la Bárány

“…Theorem [11] Let P ⊂ E d be a finite set of points in general position, in dimension d > 2. If 0 † P , then the uniform hypergraph consisting of the subsets X ∈ P d+1 such that 0 † X is tight if and only if |P | = d + 1.…”

“…More recently, two independent groups have observed that these hypotheses can be weakened a bit more; namely, it suffices that the union of pairs of colours captures the point: Theorem 4 (Arocha et al [1], Holmsen et al [11]…”

More Colourful Simplices

“…As a final remark, we mention the recent generalization of the Colourful Carathéodory Theorem in [7] and [1], in which the condition of 0 being in the convex hull of each S i is relaxed to require 0 to only be in the convex hull of S i ∪ S j for each i = j . It is natural to ask whether the minimum number of colourful simplices containing 0 is lower for configurations satisfying these weaker conditions.…”