A Poisson-Voronoi tessellation (PVT) is a tiling of the
Euclidean plane in which centers of individual tiles constitute a Poisson field
and each tile comprises the locations that are closest to a given center with
respect to a prescribed norm. Many spatial systems in which rare, randomly
distributed centers compete for space should be well approximated by a PVT.
Examples that we can handle rigorously include multitype threshold vote
automata, in which $\kappa$ different camps compete for voters stationed on
the two-dimensional lattice. According to the deterministic, discrete-time
update rule, a voter changes affiliation only to that of a unique opposing camp
having more than $\theta$ representatives in the voter's neighborhood. We
establish a PVT limit for such dynamics started from completely random
configurations, as the number of camps becomes large, so that the density of
initial "pockets of consensus" tends to 0. Our methods combine nucleation
analysis, Poisson approximation, and shape theory.