More Colourful Simplices

TL;DR

This paper proves a new lower bound on the number of simplices containing a point in the convex hull of multiple point sets in , improving previous bounds for dimensions four and higher.

Contribution

It establishes a stronger lower bound on the number of simplices containing a point in the convex hull, advancing combinatorial geometry results.

Findings

Improved lower bounds for d  when d  4

Demonstrates that any point in the convex hull is contained in at least (d+1)^2/2 simplices

Results hold for general position point sets in 

Abstract

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in \R^d is contained in at least (d+1)^2/2 simplices with one vertex from each set. This improves the known lower bounds for all d >= 4.