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.