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Published online by Cambridge University Press: 27 June 2025
We summarize the results on the differential geometric structure of Alexandrov spaces developed in [Otsu and Shioya 1994; Otsu 1995; Otsu and Tanoue a]. We discuss Riemannian and second differentiable structure and Jacobi fields on Alexandrov spaces of curvature bounded below or above.
THEOREM 1.1 [Reidemeister 1921]. Let f : ℝn → ℝ be a convex function. Then f is a.e. differentiable: more precisely, f is C1 on ℝ n \ Sf ⊂ ℝn, and the Hausdorff dimension dim# Sf = dimH SΓ is at most n — 1.
If n = 1, the map ℝ → ℝ given by u → dfu is monotone by convexity and, therefore, dfu is a.e. differentiable, that is, / is a.e. twice differentiable. In general, we have [Busemann and Feller 1935; Alexandrov 1939]:
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