  Matchbox manifolds ${\mathfrak M}$ are a special class of foliated spaces,
which includes as special examples exceptional minimal sets of foliations, weak
solenoids, suspensions of odometer and Toeplitz actions, and tiling spaces
associated to aperiodic tilings with finite local complexity. Some of these
classes of examples are endowed with an additional structure, that of a
transverse foliation, consisting of a continuous family of Cantor sets
transverse to the foliated structure. The purpose of this paper is to show that
this transverse structure can be defined on all minimal matchbox manifolds.
This follows from the construction of uniform stable Voronoi tessellations on a
dense leaf, which is the main goal of this work. From this we define a foliated
Delaunay triangulation of ${\mathfrak M}$, adapted to the dynamics of
${\mathcal F}$. The result is highly technical, but underlies the study of the
basic topological structure of matchbox manifolds in general. Our methods are
unique in that we give the construction of the Voronoi tessellations for a
complete Riemannian manifold $L$ of arbitrary dimension, with stability
estimates.
