# Basic convex analysis in metric spaces with bounded curvature

arXiv: https://arxiv.org/abs/2302.03588 · HTML: https://mlsanity.com/paper/2302.03588/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Feb 2023.

## Built on by (most cited first, 1 of 1)

- [Mean-square and sublinear convergence of a stochastic proximal point algorithm in metric spaces of nonpositive curvature](https://mlsanity.com/paper/2510.10697/) (2025)

## Abstract (arXiv)

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.