The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces
arXiv:1208.0434
Abstract
Equipped with the L^2-distortion distance, the space "X" of all metric measure spaces (X,d,m) is proven to have nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on "X" are presented.
arXiv commentNew Chapter 9 with extension to q>1, as well as with results on uniqueness of optimal couplings and existence of transport maps. Several minor corrections