Toward canonical convex functions in Alexandrov spaces

2019/10/01 by Nepechiy, Artem
#53C20 #53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1910.00253

Abstract

We construct for every finite-dimensional Alexandrov space A and every point p ∈ A a 2-convex function f_p in a small neighborhood around p, which approximates dist_p² up to second order. Moreover, the function f_p can be lifted to Gromov-Hausdorff close Alexandrov spaces of the same dimension.

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• Affine functions on Alexandrov spaces
• Quasi-convex subsets in Alexandrov spaces with lower curvature bound
• Alexandrov spaces are CS sets
• Basic convex analysis in metric spaces with bounded curvature