Geometry of measure concentration and curvature

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We studied the details of a geometric theory of metric measure spaces due to Gromov and wrote a book for it. We proved that if a sequence of metric measure spaces converges to a metric measure space with respect to the observable distance, then the curvature-dimension condition is stable. As an application, we give an estimate of the ratio of the k-th eigenvalue and the first eigenvalue of the Laplacian on a closed Riemannian manifold with nonnegative Ricci curvature, where the estimate depends only on k. Gromov defined a natural compactification of the space of metric measure spaces with the observable distance. We deeply considered it and introduce a new metric structure on it. We apply our metric structure to prove that an n-dimensional sphere of radius square root of n in a Euclidean space converges to an infinite-dimensional Gaussian space as n tends to infinity.