On a Blaschke-Santal\'o-type inequality for $r$-ball bodies

Let ${\mathbb E}^d$ denote the $d$-dimensional Euclidean space. The $r$-ball body generated by a given set in ${\mathbb E}^d$ is the intersection of balls of radius $r$ centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke-Santal\'o-type inequality for $r$-ball bodies: for all $0<k< d$ and for any set of given $d$-dimensional volume in ${\mathbb E}^d$ the $k$-th intrinsic volume of the $r$-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.