On the Number of Maximum Empty Boxes Amidst n Points

We revisit the following problem (along with its higher dimensional variant): Given a set S of n points inside an axis-parallel rectangle U in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in U but contains no points of S.(I)We prove that the number of maximum-area empty rectangles amidst n points in the plane is $$O(n \log {n} \, 2^{\alpha (n)})$$O(nlogn2α(n)), where $$\alpha (n)$$α(n) is the extremely slowly growing inverse of Ackermann’s function. The previous best bound, $$O(n^2)$$O(n2), is due to Naamad et al. (Discrete Appl Math 8(3):267–277, 1984).(II)For any $$d \ge 3$$d≥3, we prove that the number of maximum-volume empty boxes amidst n points in $$\mathbb {R}^d$$Rd is always $$O(n^d)$$O(nd) and sometimes $$\Omega (n^{\lfloor d/2 \rfloor })$$Ω(n⌊d/2⌋). This is the first superlinear lower bound derived for this problem.(III)We discuss some algorithmic aspects regarding the search for a maximum empty box in $$\mathbb {R}^3$$R3. In particular, we present an algorithm that finds a $$(1-\varepsilon )$$(1-ε)-approximation of the maximum empty box amidst n points in $$O(\varepsilon ^{-2} n^{5/3} \log ^2{n})$$O(ε-2n5/3log2n) time.