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In this note, we prove that on a surface with Alexandrov's curvature bounded below, the distance derives from a Riemannian metric whose components, for any p ∈ [1, 2), locally belong to W1,p out of a discrete singular set. This result is based on Reshetnyak's work on the more general class of surfaces with bounded integral curvature.doi:10.1515/agms-2016-0012 fatcat:xkwo6iol2rdg5lp2rhpem66yjy