Abstract
We generalize the Alexandrov–Toponogov comparison theorems to Finsler manifolds. Under suitable upper (lower, resp.) bounds on the flag and tangent curvatures together with the 2-uniform convexity (smoothness, resp.) of tangent spaces, we show the 2-uniform convexity (smoothness, resp.) of Finsler manifolds. As applications, we prove the almost everywhere existence of the second order differentials of semi-convex functions and of c-concave functions with the quadratic cost function.
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Ohta, Si. Uniform convexity and smoothness, and their applications in Finsler geometry. Math. Ann. 343, 669–699 (2009). https://doi.org/10.1007/s00208-008-0286-4
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DOI: https://doi.org/10.1007/s00208-008-0286-4