, which are centers of balls Bd(pi, ri) and Bd(qi, ri) of radius ri, for i = 1, …, N. In  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 , and for the case when the centers are contracted continuously for all d in . One extension
of the Kneser-Poulsen conjecture, suggested in , 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