• Corpus ID: 16927657

Optimal Transport and Tessellation

  title={Optimal Transport and Tessellation},
  author={Martin Huesmann},
  journal={arXiv: Probability},
  • M. Huesmann
  • Published 4 August 2009
  • Mathematics
  • arXiv: Probability
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… 

Figures from this paper



Continuity, curvature, and the general covariance of optimal transportation

Let M and M be n-dimensional manifolds equipped with suitable Borel probability measures ρ and ρ. For subdomains M and M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation

Random Laguerre tessellations

A systematic study of random Laguerre tessellations, weighted generalisations of the well-known Voronoi tessellations, is presented. We prove that every normal tessellation with convex cells in

On the regularity of solutions of optimal transportation problems

We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition

Polar factorization of maps on Riemannian manifolds

Abstract. Let (M,g) be a connected compact manifold, C3 smooth and without boundary, equipped with a Riemannian distance d(x,y). If $ s : M \to M $ is merely Borel and never maps positive volume

Minkowski-type theorems and least-squares partitioning

If the authors fix a set X of m points in the plane, this set is partitioned by the Voronoi diagram of S into subsets, which defines an assignment function A : X + S, given by the diagram.

Entropic Measure on Multidimensional Spaces

We construct the entropic measure \(\mathbb{P}^\beta\) on compact manifolds of any dimension. It is defined as the push forward of the Dirichlet process (a random probability measure, well-known to

On Optimal Multivariate Couplings

As consequence of a characterization of optimal multivariate coupling (transportation) problems we obtain the existence of optimal Monge solutions as well as an explicit construction method for

Regularity of Potential Functions of the Optimal Transportation Problem

The potential function of the optimal transportation problem satisfies a partial differential equation of Monge-Ampère type. In this paper we introduce the notion of a generalized solution, and prove

The geometry of optimal transportation

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 1. Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . 120 2. Background on optimal

Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften 338

  • 2009