30 jan 2013 -- 17:00 [open in google calendar]
Sala Seminari, Department of Mathematics, Pisa University
We give a new notion of angle in general metric spaces; more precisely, given a triple of points p,x,q in a metric space (X,d), we introduce the notion of angle cone as being an interval 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. We show that in general the angle cone is not single valued even in case the metric space is a smooth Riemannian manifold), 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 fimensional 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 in X, the angle cone is single valued for m-a.e. x in 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.