  The Kneser-Poulsen conjecture says that if a finite collection of balls in a
d-dimensional Euclidean space 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.
