Let $N$ balls of the same radius be given in a $d$-dimensional real normed vector space, i.e., in a Minkowski $d$-space. Then apply a uniform contraction to the centers of the $N$ balls without changing the common radius. Here a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. The main results of this paper state that a uniform contraction of the centers does not increase (resp., decrease) the volume of the union (resp., intersection) of $N$ balls in Minkowski $d$-space, provided that $N\geq 2^d$ (resp., $N\geq 3^d$ and the unit ball of the Minkowski $d$-space is a generating set). Some improvements are presented in Euclidean spaces.

中文翻译：

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明可夫斯基空间中球的均匀收缩

让相同半径的 $N$ 个球在 $d$ 维实范数向量空间中给出，即在 Minkowski $d$ 空间中。然后在不改变公共半径的情况下对 $N$ 球的中心应用均匀收缩。这里的均匀收缩是一种收缩，其中第一组中心的所有成对距离都大于第二组中心的所有成对距离。本文的主要结果表明，中心的均匀收缩不会增加（相应地，减少）Minkowski $d$-空间中 $N$ 球的并集（相应地，交集）的体积，前提是 $ N\geq 2^d$（相应地，$N\geq 3^d$ 和 Minkowski $d$-空间的单位球是一个生成集）。在欧几里得空间中提出了一些改进。