Homogeneous measures and positive Alexandrov curvature (Q1983994)

Homogeneous probability measures are important in the theory of turbulence of Kolmogorov. Moreover, the spaces of all probability measures on Hilbert spaces, equipped with a Wasserstein distance, become a positive curvature space in the sense of A. D. Alexandrov. This paper is an attempt to describe the geometry of the subspace of homogeneous probability measures in terms of the Wasserstein metric. The main result states that, on appropriate Hilbert spaces, the geodesics joining homogeneous measures stay in the space of homogeneous measures. As a consequence, the authors obtain that the space of homogeneous measures is a metric space of positive curvature in the sense of Alexandrov.