Asymptotic quantization for probability measures on Riemannian manifolds

¹ University of Rome Sapienza, Department of Mathematics Guido Castelnuovo, Piazzale Aldo Moro 5, 00185 Rome, Italy.
iacobelli@mat.uniroma1.it
² Ecole Polytechnique, Centre de mathématiques Laurent Schwartz, 91128 Palaiseau cedex, France.

Received: 4 December 2014

Abstract

In this paper we study the quantization problem for probability measures on Riemannian manifolds. Under a suitable assumption on the growth at infinity of the measure we find asymptotic estimates for the quantization error, generalizing the results on R^d. Our growth assumption depends on the curvature of the manifold and reduces, in the flat case, to a moment condition. We also build an example showing that our hypothesis is sharp.