Subdifferential theory and the Fenchel conjugate via Busemann functions on Hadamard manifolds
G. C. Bento, J. X. Cruz Neto, I. D. L. Melo
TL;DR
The paper develops a subdifferential theory on Hadamard manifolds via Busemann functions and establishes an equality condition for the Fenchel-Young inequality, linking a primal function to its Riemannian Fenchel conjugate. It extends affine-function rigidity results to contexts with Ricci curvature, showing that nontrivial affine functions imply a zero Ricci direction and a product-like structure, and it provides curvature-dependent Fenchel-Moreau-type bounds. The authors also analyze the conjugate structure under isometries, derive radial-function subdifferential characterizations, and introduce quantitative measures of nonlinearity that capture curvature effects on biconjugation. These results deliver an intrinsic duality framework for convex analysis on Hadamard manifolds and pave the way for curvature-aware optimization methods in non-Euclidean settings.
Abstract
In this paper, we propose a notion of subdifferential defined via Busemann functions and use it to identify a condition under which the Fenchel-Young inequality of Bento, Cruz Neto and Melo (Appl. Math. Optim. 88:83, 2023) holds with equality. This equality condition is particularly significant, as it captures a fundamental duality principle in convex analysis, linking a primal convex function to its conjugate and clarifying the sharpness of the associated inequality on Riemannian manifolds. We also investigate the existence of non-trivial affine functions under Ricci curvature information. In particular, we extend the result of Bento, Cruz Neto and Melo, originally formulated for the case of negative Ricci curvature on an open set, to manifolds whose Ricci curvature may be non-zero. As a consequence, we prove new non-existence criteria for non-trivial affine functions and show that the assumption of non-zero Ricci curvature is, in general, necessary to ensure such a rigidity conclusion.