Covering a Unit Hypercube with Hypercubes

Abstract

The covering and packing of a unit square (resp. cube) with squares (resp. cubes) are considered. In \(d\)-dimensional Euclidean space \(\mathbb{E}^d\), the size of a \(d\)-hypercube is given by its side length and the size of a covering is the total size of the \(d\)-hypercubes used to cover the unit hypercube. Denote by \(g_d(n)\) the smallest size of a minimal covering (consisting of \(n\) hypercubes) of a \(d\)-dimensional unit hypercube. In this paper, we consider the problem of covering a unit hypercube with hypercubes in \(\mathbb{E}^d\) for \(d \geq 4\) and determine the tight upper bound and lower bound for \(g_d(n)\).