Volumetric bounds for intersections of congruent balls

TL;DR

This paper establishes volumetric bounds for intersections of congruent balls in Euclidean space, identifying extremal shapes like $r$-lenses and $r$-spindles, and extends previous results on $r$-ball bodies.

Contribution

It provides new bounds on intrinsic volumes of $r$-ball bodies based on inradius and circumradius, extending earlier volumetric estimates.

Findings

$r$-lenses have minimal inradius among fixed-volume $r$-ball bodies.

$r$-spindles have maximal circumradius among fixed-volume $r$-ball bodies.

The results generalize and improve previous volumetric bounds for $r$-ball bodies.

Abstract

We investigate the intersections of balls of radius $r$, called $r$-ball bodies, in Euclidean $d$-space. An $r$-lense (resp., $r$-spindle) is the intersection of two balls of radius $r$ (resp., balls of radius $r$ containing a given pair of points). We prove that among $r$-ball bodies of given volume, the $r$-lense (resp., $r$-spindle) has the smallest inradius (resp., largest circumradius). In general, we upper (resp., lower) bound the intrinsic volumes of $r$-ball bodies of given inradius (resp., circumradius). This complements and extends some earlier results on volumetric estimates for $r$-ball bodies.