Optimal Transport and Tessellation

TL;DR

This paper explores how optimal transport can generate tessellations of Riemannian manifolds into regions of specified sizes, providing geometric descriptions for various cost functions including Laguerre and Johnson Mehl tessellations.

Contribution

It offers a unified geometric framework for tessellations derived from optimal transport with different cost functions on Riemannian manifolds.

Findings

Provides geometric descriptions for all p in the cost function family

Specializes to Laguerre tessellations for p=2

Derives Johnson Mehl diagrams for p=1 on Riemannian manifolds

Abstract

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.