On the size of the largest empty box amidst a point set

TL;DR

This paper investigates the size of the largest empty axis-aligned box in a point set within a hypercube, showing that its volume can grow with dimension, roughly proportional to d log d, countering previous asymptotic results.

Contribution

It proves that the largest empty box volume increases with dimension, providing a lower bound proportional to d log d, which was not previously established.

Findings

Largest empty box volume grows with dimension

Lower bound of volume is proportional to d log d

Contrasts with known asymptotic order 1/n for fixed d

Abstract

The problem of finding the largest empty axis-parallel box amidst a point configuration is a classical problem in computational geometry. It is known that the volume of the largest empty box is of asymptotic order $1/n$ for $n\to\infty$ and fixed dimension $d$. However, it is natural to assume that the volume of the largest empty box increases as $d$ gets larger. In the present paper we prove that this actually is the case: for every set of $n$ points in $[0, 1]^d$ there exists an empty box of volume at least $c_d n^{-1}$ , where $c_d \to \infty$ as $d\to \infty$. More precisely, $c_d$ is at least of order roughly $\log d$.