Upper bounds for Heilbronn's triangle problem in higher dimensions

TL;DR

This paper introduces a new approach to establish upper bounds for generalized Heilbronn's triangle problem in higher dimensions, providing bounds on volumes of simplices and convex hulls formed by points in high-dimensional spaces.

Contribution

The paper presents a novel, simplified method to derive upper bounds for volume-related problems in higher-dimensional Heilbronn's triangle generalizations.

Findings

Existence of small-volume simplices among point sets in high dimensions.

Bounds on convex hull volumes for subsets of points.

Explicit volume bounds for large point subsets spanning simplices.

Abstract

We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$ contains - $d+1$ points which span a simplex of volume at most $C_d n^{-\log d+ 6}$, - $1.1 d$ points whose convex hull has volume at most $C_d n^{-1.1}$, - $k\ge 4\sqrt{d}$ points which span a $(k-1)$-dimensional simplex of volume at most $C_d n^{-\frac{k-1}{d} - \frac{k^2}{8d^2}}$.