• Corpus ID: 16927657

# Optimal Transport and Tessellation

@article{Huesmann2009OptimalTA,
title={Optimal Transport and Tessellation},
author={Martin Huesmann},
journal={arXiv: Probability},
year={2009}
}
• 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…

## References

SHOWING 1-10 OF 15 REFERENCES

### Continuity, curvature, and the general covariance of optimal transportation

• Mathematics
• 2007
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

• Mathematics
Advances in Applied Probability
• 2008
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

• Mathematics
SCG '92
• 1992
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

• Mathematics
• 1997
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

• Mathematics
• 2005
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

• Economics
• 1996
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 1. Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . 120 2. Background on optimal

• 2009