On the weighted Kneser-Poulsen conjecture

Abstract

Suppose that p = (p1, p2, …, pN) and q = (q1, q2, …, qN) are two configurations in $\mathbb{E}^d$ , which are centers of balls B^d(p _i , r _i ) and B^d(q _i , r _i ) of radius r _i , for i = 1, …, N. In [9] it was conjectured that if the pairwise distances between ball centers p are contracted in going to the centers q, then the volume of the union of the balls does not increase. For d = 2 this was proved in [1], and for the case when the centers are contracted continuously for all d in [2]. One extension of the Kneser-Poulsen conjecture, suggested in [6], was to consider various Boolean expressions in the unions and intersections of the balls, called flowers, where appropriate pairs of centers are only permitted to increase, and others are only permitted to decrease. Again under these distance constraints, the volume of the flower was conjectured to change in a monotone way.

Here we show that these generalized Kneser-Poulsen flower conjectures are equivalent to an inequality between certain integrals of functions (called flower weight functions) over $\mathbb{E}^d$ , where the functions in question are constructed from maximum and minimum operations applied to functions each being radially symmetric monotone decreasing and integrable.