A new lower bound based on Gromov's method of selecting heavily covered points

TL;DR

This paper introduces a new lower bound for the number of simplices containing a point in a set of points in R^d, improving previous bounds using a combination of topological and combinatorial methods.

Contribution

It presents a novel lower bound on c_d by enhancing Gromov's topological approach with extremal combinatorics techniques, applicable for all dimensions.

Findings

Improved lower bound on c_3 from 0.06332 to over 0.07480

Enhanced lower bounds for c_d in arbitrary dimensions

Bridged the gap between known bounds and upper bounds for c_3

Abstract

Boros and Furedi (for d=2) and Barany (for abritrary d) proved that there exists a positive real number c_d such that for every set P of n points in R^d in general position, there exists a point of R^d contained in at least c_d n!/(d+1)!(n-d-1)! d-simplices with vertices at the points of P. Gromov improved the lower bound on c_d by topological means. Using methods from extremal combinatorics, we improve one of the quantities appearing in Gromov's approach and thereby provide a new stronger lower bound on c_d for arbitrary d. In particular, we improve the lower bound on c_3 from 0.06332 to more than 0.07480; the best upper bound known on c_3 being 0.09375.