Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces

Equipped with the L2,q-distortion distance 𝚫2,q, the space 𝕏2,q 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 ̅𝕏2,q are presented.