We revisit a classical problem in computational geometry: finding the
largest-volume axis-aligned empty box (inside a given bounding box) amidst n
given points in d dimensions. Previously, the best algorithms known have
running time O(nlog2n) for d=2 (by Aggarwal and Suri [SoCG'87]) and near
nd for d≥3. We describe faster algorithms with running time (i)
O(n2O(log∗n)logn) for d=2, (ii) O(n2.5+o(1)) time for d=3,
and (iii) O(n(5d+2)/6) time for any constant d≥4.
To obtain the higher-dimensional result, we adapt and extend previous
techniques for Klee's measure problem to optimize certain objective functions
over the complement of a union of orthants.Comment: full version of a SoCG 2021 pape