Fenchel Duality and a Separation Theorem on Hadamard Manifolds
Abstract
In this paper, we introduce a definition of Fenchel conjugate and Fenchel biconjugate on Hadamard manifolds based on the tangent bundle. Our definition overcomes the inconvenience that the conjugate depends on the choice of a certain point on the manifold, as previous definitions required. On the other hand, this new definition still possesses properties known to hold in the Euclidean case. It even yields a broader interpretation of the Fenchel conjugate in the Euclidean case itself. Most prominently, our definition of the Fenchel conjugate provides a FenchelMoreau Theorem for geodesically convex, proper, lower semicontinuous functions. In addition, this framework allows us to develop a theory of separation of convex sets on Hadamard manifolds, and a strict separation theorem is obtained.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 DOI:
 10.48550/arXiv.2102.11155
 arXiv:
 arXiv:2102.11155
 Bibcode:
 2021arXiv210211155S
 Keywords:

 Mathematics  Optimization and Control;
 Mathematics  Differential Geometry
 EPrint:
 doi:10.1137/21M1400699