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Upper bounds for Heilbronn's triangle problem in higher dimensions

Dmitrii Zakharov

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 $C_d n^{-\frac{k-1}{d} - \frac{k^2}{8d^2}}$.

Upper bounds for Heilbronn's triangle problem in higher dimensions

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 , any subset of of size contains - points which span a simplex of volume at most , - points whose convex hull has volume at most , - points which span a -dimensional simplex of volume at most .
Paper Structure (6 sections, 12 theorems, 45 equations)

This paper contains 6 sections, 12 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Proof of Theorem \ref{['main']}
  4. Proofs of corollaries
  5. Simplices with faces of small volume
  6. Proof of Theorem \ref{['largek']} and Proposition \ref{['smalld']}

Key Result

Theorem 1.1

Fix $2 \leqslant k \leqslant d+1$ and $1 \leqslant \ell < k$. Then for some constant $C = C(d)$ and all $n' > C n$ we have:

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • ...and 2 more