The Space of Spaces: Curvature Bounds and Gradient Flows on the Space ofric Measure Spaces Peter Lunt Rose Cherie Reissman and Mark

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The Space of Spaces: Curvature Bounds and Gradient Flows on the Space ofric Measure Spaces Peter Lunt Rose Cherie Reissman and MarkEquipped with the L2,q distortion distance 2,q, the space 2,q of all metric measure spaces (X, ?, ) 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 ?2,q are presented.