Kneser-Poulsen conjecture for a small number of intersections
The Kneser-Poulsen conjecture says that if a finite collection of balls in the Euclidean space $E^d$ is rearranged so that the distance between each pair of centers does not get smaller, then the volume of the union of these balls also does not get smaller. In this paper, we prove that if in the initial configuration the intersection of any two balls has common points with no more than $d+1$ other balls, then the conjecture holds.
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