Fenchel Duality and a Separation Theorem on Hadamard Manifolds

TL;DR

This paper extends Fenchel duality and separation theorems to Hadamard manifolds, providing a new conjugate definition that is independent of point choice and retains Euclidean properties.

Contribution

It introduces a point-independent Fenchel conjugate on Hadamard manifolds, generalizing Euclidean duality and establishing a separation theorem for convex sets.

Findings

A new Fenchel conjugate definition on Hadamard manifolds that is point-independent.

A Fenchel-Moreau Theorem for geodesically convex functions on manifolds.

A strict separation theorem for convex sets 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.