Inventiones mathematicae

, Volume 146, Issue 2, pp 219–257

A Riemannian interpolation inequality à la Borell, Brascamp and Lieb

  • Dario Cordero-Erausquin
  • Robert J. McCann
  • Michael Schmuckenschläger

DOI: 10.1007/s002220100160

Cite this article as:
Cordero-Erausquin, D., McCann, R. & Schmuckenschläger, M. Invent. math. (2001) 146: 219. doi:10.1007/s002220100160


A concavity estimate is derived for interpolations between L1(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian versions of these theorems incorporate a volume distortion factor which can, however, be controlled via lower bounds on Ricci curvature. The method uses optimal mappings from mass transportation theory. Along the way, several new properties are established for optimal mass transport and interpolating maps on a Riemannian manifold.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Dario Cordero-Erausquin
    • 1
  • Robert J. McCann
    • 2
  • Michael Schmuckenschläger
    • 3
  1. 1.Equipe d'Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée, 77454 Marne-la-Vallée Cedex 2, France (e-mail:
  2. 2.Department of Mathematics, University of Toronto, Toronto Ontario Canada M5S 3G3 (e-mail:
  3. 3.Institut für Analysis und Numerik, Universität Linz, A-4040 Linz, Österreich (e-mail: