Given a compactly supported probability measure on a Riemannian manifold, we study the asymptotic speed at which it can be approximated (in Wasserstein distance of any exponent p) by finitely supported measure. This question has been studied under the names of ``quantization of distributions'' and, when p=1, ``location problem''. When p=2, it is linked with Centroidal Voronoi Tessellations.
- Pub Date:
- March 2010
- Mathematics - Optimization and Control;
- Mathematics - Functional Analysis
- v2: the main result is extended to measures defined on a manifold. v3: references added. 25 pp. To appear in ESAIM:COCV