  Optimal transport from the volume measure to a convex combination of Dirac
measures yields a tessellation of a Riemannian manifold into pieces of
arbitrary relative size. This tessellation is studied for the cost functions
$c_p(z,y)=\frac{1}{p}d^p(z,y)$ and $1\leq p<\infty$. Geometric descriptions of
the tessellations for all $p$ is obtained for compact subsets of the Euclidean
space. For $p=2$ this approach yields Laguerre tessellations. For $p=1$ it
induces Johnson Mehl diagrams for all compact Riemannian manifolds.
