Basic convex analysis in metric spaces with bounded curvature

TL;DR

This paper extends fundamental convex analysis concepts to metric spaces with bounded curvature, specifically Alexandrov spaces, by defining subgradients, normal cones, and optimality conditions without relying on differentiable structures.

Contribution

It introduces a framework for convex analysis in Alexandrov spaces with curvature bounds, including subgradients, normal cones, and optimality conditions, broadening the scope beyond smooth manifolds.

Findings

Existence of subgradients via projection and normal cones

Relation of subgradients to classical affine minorant property

Necessary optimality condition for sum of convex functions

Abstract

Differentiable structure ensures that many of the basics of classical convex analysis extend naturally from Euclidean space to Riemannian manifolds. Without such structure, however, extensions are more challenging. Nonetheless, in Alexandrov spaces with curvature bounded above (but possibly positive), we develop several basic building blocks. We define subgradients via projection and the normal cone, prove their existence, and relate them to the classical affine minorant property. Then, in what amounts to a simple calculus or duality result, we develop a necessary optimality condition for minimizing the sum of two convex functions.