  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. In this paper we prove
the following Blaschke-Santal\'o-type inequalities for $r$-ball bodies: for all
$1\leq k\leq d$ and for any set of given 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. As an application we investigate the Gromov-Klee-Wagon
problem for congruent balls in ${\mathbb E}^d$, which is a question on proving
or disproving that if the centers of a family of $N$ congruent balls in
${\mathbb E}^d$ are contracted, then the volume of the intersection does not
decrease. In particular, we investigate this problem for uniform contractions,
which are contractions where all the pairwise distances in the first set of
centers are larger than all the pairwise distances in the second set of
centers, that is, when the pairwise distances of the two sets are separated by
some positive real number. The author and M. Nasz\'odi [Discrete Comput. Geom.
60/4 (2018), 967-980] proved that the intrinsic volumes of the intersection of
$N$ congruent balls in ${\mathbb E}^d$, $d>1$ increase under any uniform
contraction of the center points when $ N\geq \left(1+\sqrt{2}\right)^d$. We
give a short proof of this result using the Blaschke-Santal\'o-type
inequalities of $r$-ball bodies and improve it for $d\geq 42$.
