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arXiv · 2102.11155

Fenchel Duality and a Separation Theorem on Hadamard Manifolds

Maurício Silva Louzeiro·Ronny Bergmann·Roland Herzog
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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.

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Maurício Silva Louzeiro, Ronny Bergmann, Roland Herzog. 2021-02-22. Fenchel Duality and a Separation Theorem on Hadamard Manifolds. https://doi.org/10.1137/21m1400699

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