TY - JOUR

T1 - On the size of the largest empty box amidst a point set

AU - Aistleitner, Christoph

AU - Hinrichs, Aicke

AU - Rudolf, Daniel

PY - 2017

Y1 - 2017

N2 - 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 → ∞ 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 cdn−1, where cd → ∞ as d → ∞. More precisely, cd is at least of order roughly log d.

AB - 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 → ∞ 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 cdn−1, where cd → ∞ as d → ∞. More precisely, cd is at least of order roughly log d.

U2 - 10.1016/j.dam.2017.06.008

DO - 10.1016/j.dam.2017.06.008

M3 - Article

SN - 1872-6771

VL - 230

SP - 146

EP - 150

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

ER -