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 Fenchel-Moreau 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.
Keywords
Cite
@article{arxiv.2102.11155,
title = {Fenchel Duality and a Separation Theorem on Hadamard Manifolds},
author = {Maurício Silva Louzeiro and Ronny Bergmann and Roland Herzog},
journal= {arXiv preprint arXiv:2102.11155},
year = {2022}
}