On the Volume of the Union of Balls

We prove that if some balls in the Euclidean space move continuously in such a way that the distances between their centers decrease, then the volume of their union cannot increase. The proof is based on a formula expressing the derivative of the volume of the union as a linear combination of the derivatives of the distances between the centers with nonnegative coefficients. Received September 6, 1996, and in revised form March 26, 1997.