Optimal Transport and Tessellation

Mathematics – Probability

Scientific paper

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11 pages

Scientific paper

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 and the sphere. For $p=1$ this approach yields Laguerre tessellations for all compact Riemannian manifolds.

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