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

TL;DR

This paper introduces a novel way to define an angle between three points in a metric space, generalizing classical angles and showing that under certain conditions, the angle is well-defined almost everywhere in metric measure spaces.

Contribution

It proposes a new notion of angle cone in metric spaces, extending classical concepts and proving its properties and consistency with existing geometric constructions.

Findings

The angle cone coincides with standard angles in Euclidean and Riemannian settings.

In general metric spaces, the angle cone is not always single-valued, but is almost everywhere single-valued in metric measure spaces.

The definition is consistent with recent Gromov-Hausdorff limit constructions in Riemannian geometry.

Abstract

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 ${\angle_{pxq}}$ as being an interval ${\angle_{pxq}}:=[\angle^-_{pxq},\angle^+_{pxq}]$, where the quantities $\angle^\pm_{pxq}$ 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 $p,q \in X$, the angle cone at $x$ 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.