A new notion of angle between three points in a metric space

Abstract

Abstract We give a new notion of angle in general metric spaces; more precisely, given three points p,x,q in a metric space (X,d), we introduce the notion of angle cone ∠ 𝐩𝐱𝐪 ${{\angle _{\mathbf {pxq}}}}$ as being an interval ∠ 𝐩𝐱𝐪 : = [ ∠ p x q - , ∠ p x q + ] ${{\angle _{\mathbf {pxq}}}:=[\angle ^-_{pxq},\angle ^+_{pxq}]}$ , where the quantities ∠ p x q ± ${\angle ^\pm _{pxq}}$ are defined in terms of the distance functions from p and q via a duality construction of differentials and gradients holding for locally Lipschitz functions on a general metric space. Our definition in the Euclidean plane gives the standard angle between three points and in a Riemannian manifold coincides with the usual angle between the geodesics if x is not in the cut locus of p or q. We show that in general the angle cone is not single-valued (even in case the metric space is a smooth Riemannian manifold, if x is in the cut locus of p or q), but if we endow the metric space with a positive Borel measure 𝔪 obtaining the metric measure space (X,d,𝔪), then under quite general assumptions (which include many fundamental examples as Riemannian manifolds, finite dimensional Alexandrov spaces with curvature bounded from below, Gromov–Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below, and normed spaces with strictly convex norm), fixed p,q ∈ X, the angle cone at x is single-valued for 𝔪-a.e. x ∈ X. We prove some basic properties of the angle cone (such as the invariance under homotheties of the space) and we analyze in detail the case (X,d,𝔪) is a measured-Gromov–Hausdorff limit of a sequence of Riemannian manifolds with Ricci curvature bounded from below, showing the consistency of our definition with a recent construction of Honda [“A weakly second order differential structure on rectifiable metric measure spaces”, preprint 2012].