On the Volume of Flowers in Space Forms

Let f(x ₁,..., x _N ) be a lattice polynomial in N variables, in which each variable occurs exactly once, B ₁,..., B _N be smoothly moving balls in the hyperbolic, Euclidean, or spherical space. Introducing a suitable modification of the Dirichlet–Voronoi decomposition, we prove a formula for the derivative of the volume of the domain f(B ₁,..., B _N ). As an application of the formula, we show that the volume increases if the balls move continuously in such a way that the functions _ij d _ij increase for all 1 i < j N, where _ij is a sign depending on f, d _ij is the distance between the centers of B _I and B _j .