Basic convex analysis in metric spaces with bounded curvature. (English) Zbl 1539.53054
Summary: 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.
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 53C45 | Global surface theory (convex surfaces à la A. D. Aleksandrov) |
| 52A07 | Convex sets in topological vector spaces (aspects of convex geometry) |
| 46A03 | General theory of locally convex spaces |
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