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

We give a new notion of angle in general metric spaces; more precisely, given a triple a points p, x, q in a metric space (X, d), we introduce the notion of angle cone ∠pxq as being an interval ∠pxq := [∠pxq,∠ + pxq], where the quantities ∠ ± 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 [24].