On the volume of the union of balls (Q1275668)

A longstanding and exasperating conjecture of Poulsen and Kneser states that if a finite collection of balls in \(\mathbb{R}^d\) is moved so that each pair of centers ends up closer than before, the volume of their union cannot thereby be increased. Bollobás proved in 1968 that the conjecture is true for congruent disks in the plane, if the disks are moved so that the distance between each pair of centers is always nonincreasing; and this has since been generalized to noncongruent disks. The present paper extends these results to balls in higher-dimensional spaces, while retaining the condition that distances must never increase. The author derives a formula for the derivative of the volume as a function of the derivatives of the various distances between centers. This immediately yields a proof for differentiable motions. However, it is possible that, while the distances between centers are well-behaved, the paths of individual centers are much less so. Therefore, the result does not extend trivially to arbitrary continuous motions; it may be necessary, in the author's words, to ``calm down a nonrectifiable `shivering' motion of the centers''. It is shown -- via an excursion into compact Lie groups -- that this can be done, and the main result follows.