On the Volume of Boolean expressions of Large Congruent Balls

TL;DR

This paper investigates the asymptotic volume of Boolean expressions of large congruent balls in high-dimensional space, revealing how the coefficients relate to generalized intrinsic volumes and extending known geometric inequalities.

Contribution

It introduces a polynomial approximation for the volume of Boolean ball expressions and generalizes intrinsic volumes and mean width to this context.

Findings

Volume approximated by a polynomial with error term $O(r^{d-3})

Top coefficients relate to generalized intrinsic volumes

Extended monotonicity of mean width to Boolean analogues

Abstract

We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the $d$-dimensional Euclidean space. When the radius $r$ of the balls is large, this volume can be approximated by a polynomial of $r$, which will be computed up to an $O(r^{d-3})$ error term. We study how the top coefficients of this polynomial depend on the set of the centers. It is known that in the case of the union of the balls, the top coefficients are some constant multiples of the intrinsic volumes of the convex hull of the centers. Thus, the coefficients in the general case lead to generalizations of the intrinsic volumes, in particular, to a generalization of the mean width of a set. Some known results on the mean width, along with the theorem on its monotonicity under contractions are extended to the "Boolean analogues" of the mean width.