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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). A 'uniform contraction' is a contraction where all the pairwise distances in the first set of points are larger than all the pairwise distances in the second set of points. We show that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that $N\geq(1+\sqrt{2})^d$. Joint work with K\'aroly Bezdek.