We examine the geometry of the subspace of homogeneous probability measures in terms of the 2-Wasserstein metric on the space of all probability measures on Hilbert spaces of functions. We show that, on appropriate Hilbert spaces, the geodesics joining homogeneous measures stay in the space of homogeneous measures and, as a result, the homogeneous measures themselves form a space of nonpositive curvature in the sense of A. D. Alexandrov.
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Translated from Problemy Matematicheskogo Analiza 111, 2021, pp. 91-98.
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Dostoglou, S., Kahl, J.D. Homogeneous Measures and Positive Alexandrov Curvature. J Math Sci 257, 652–661 (2021). https://doi.org/10.1007/s10958-021-05507-y
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DOI: https://doi.org/10.1007/s10958-021-05507-y