特殊压缩的Kneser-Poulsen猜想
The Kneser--Poulsen conjecture for special contractions
摘要：
Kneser-Poulsen猜想指出，如果${\mathbb E}^d$中的$N$单位球族的中心收缩，则并集的体积（分别为，交叉点）不增加（相应地，减少）。我们考虑两种类型的特殊收缩。首先，\emph{uniform contraction}是第一组中心中的所有成对距离大于第二组中心中的所有成对距离的收缩。我们得到，一个统一的收缩中心不减少的体积的交叉点的球，提供$N\geq(1+\sqrt{2})^d$。我们的结果扩展到内禀体积。我们证明了一个类似的结果有关的体积联盟。其次，\emph{strong contraction}是每个坐标中的收缩。我们证明了这个猜想对强压缩也成立。事实上，这个结果可以推广到任意的无条件物体，而不是球。
英文摘要：
The Kneser--Poulsen Conjecture states that if the centers of a family of $N$ unit balls in ${\mathbb E}^d$ is contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of special contractions. First, a \emph{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. We obtain that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that $N\geq(1+\sqrt{2})^d$. Our result extends to intrinsic volumes. We prove a similar result concerning the volume of the union. Second, a \emph{strong contraction} is a contraction in each coordinate. We show that the conjecture holds for strong contractions. In fact, the result extends to arbitrary unconditional bodies in the place of balls.
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