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–Santaló-type inequalities for $r$-ball bodies: for all $1\le k\le d$ 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. Bezdek and Naszódi (2018), 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 $N\ge {\left(1+\sqrt{2}\right)}^d$. We give a short proof of this result using the Blaschke–Santaló-type inequalities of $r$-ball bodies and improve it for $d\ge 42$.

让 ${\mathbb{Ë}}^d$ 表示 $d$维欧氏空间。的$\mathrm{[R}$给定集合生成的球体 ${\mathbb{Ë}}^d$ 是半径球的交点 $\mathrm{[R}$以给定集合的点为中心。在本文中，我们证明了以下Blaschke-Santaló型不等式$\mathrm{[R}$球体：所有人 $\mathrm{1个}\le ķ\le d$ 对于任何给定的体积 ${\mathbb{Ë}}^d$ 的 $ķ$的固有体积 $\mathrm{[R}$如果集合是球，则集合生成的-ball body变为最大。作为一个应用程序，我们研究了球在等距中的Gromov–Klee–Wagon问题。${\mathbb{Ë}}^d$，这是关于证明或否定一个家庭的中心 $ñ$ 一致的球 ${\mathbb{Ë}}^d$收缩，则交点的体积不会减少。特别地，我们研究均匀收缩的问题，即第一组中心的所有成对距离均大于第二组中心的所有成对距离（即当两组的成对距离时）被一些正实数分开。Bezdek和Naszódi（2018）证明了交点的内在体积$ñ$ 一致的球 ${\mathbb{Ë}}^d$， $d>\mathrm{1个}$ 在中心点均匀收缩时增加 $ñ\ge {\left(\mathrm{1个}+\sqrt{2}\right)}^d$。我们使用Blaschke-Santaló型不等式给出了该结果的简短证明$\mathrm{[R}$球体并对其进行改进 $d\ge 42$。