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We give the first nontrivial upper and lower bounds on the maximum volume ofan empty axis-parallel box inside an axis-parallel unit hypercube in $\RR^d$containing $n$ points. For a fixed $d$, we show that the maximum volume is ofthe order $\Theta(\frac{1}{n})$. We then use the fact that the maximum volumeis $\Omega(\frac{1}{n})$ in our design of the first efficient$(1-\eps)$-approximation algorithm for the following problem: Given anaxis-parallel $d$-dimensional box $R$ in $\RR^d$ containing $n$ points, computea maximum-volume empty axis-parallel $d$-dimensional box contained in $R$. Therunning time of our algorithm is nearly linear in $n$, for small $d$, andincreases only by an $O(\log{n})$ factor when one goes up one dimension. Noprevious efficient exact or approximation algorithms were known for thisproblem for $d \geq 4$. As the problem has been recently shown to be NP-hard inarbitrary high dimensions (i.e., when $d$ is part of the input), the existenceof efficient exact algorithms is unlikely.  We also obtain tight estimates on the maximum volume of an emptyaxis-parallel hypercube inside an axis-parallel unit hypercube in $\RR^d$containing $n$ points. For a fixed $d$, this maximum volume is of the sameorder order $\Theta(\frac{1}{n})$. A faster $(1-\eps)$-approximation algorithm,with a milder dependence on $d$ in the running time, is obtained in this case.

On the largest empty axis-parallel box amidst $n$ points