On the Volume of Unions and Intersections of Balls in Euclidean Space

Abstract

We study the following old problem: Given two sequences { _ai } ^N _i =1 and {b _i } ^N _i =1 of N points in ℝⁿ, and positive scalars {r _i } ^N _i =1 such that |a _i – a _j | ≤ |b _i – b _j | for all i, j, does it follow that

$$ vo{l_n}\left({\bigcup\limits_{i = 1}^N {B\left({{a_i},{r_i}} \right)}} \right) \leqslant vo{l_n}\left({\bigcup\limits_{i = 1}^N {B\left({{b_i},{r_i}} \right)}} \right) $$

where |. | is the Euclidean norm and B(a, r) is the ball centered at a and of radius r? Under some additional assumptions, we give a probabilistic proof of this and of other related results.