Volumetric bounds for intersections of congruent balls

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.