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Fenchel Conjugate via Busemann Function on Hadamard Manifolds

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Abstract

In this paper we introduce a Fenchel-type conjugate, given as the supremum of convex functions, via Busemann functions. It is known that Busemann functions are smooth convex functions with constant norm gradient. Our study ensures that our proposal on Fenchel conjugate is the most adequate to cover the absence of approximations by non-constant affine functions on Hadamard manifolds. More precisely, as a first contribution of the paper we prove that any affine function is constant in a complete and connected Riemannian manifold of nonpositive Ricci curvature on some open set. Moreover, we show the influence of the sectional curvature in obtaining the main results. In particular, we illustrate that the difference between a proper, lsc, convex function and its biconjugate is a constant that depends on the sectional curvature of the manifold, which show that a Fenchel-Moreau-type theorem is directly influenced by the sectional curvature in general. We also present some applications formulated in terms of the Fenchel conjugate.

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Funding

This work was funded by the CNPq Grants 314106/2020-0, 302156/2022-4, FAPEPI.

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Authors and Affiliations

  1. IME - Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia-GO, 74001-970, Brazil

    Glaydston de C. Bento

  2. DM - Departamento de Matemática, Universidade Federal do Piauí, Teresina, PI, 64049-550, Brazil

    João Cruz Neto & Ítalo Dowell L. Melo

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  1. Glaydston de C. Bento
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All authors contributed to the study conception and design, actively participated in the construction of the manuscript and approved the final version.

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This work was funded by the CNPq Grants 314106/2020-0, 302156/2022-4, FAPEPI.

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Bento, G.d.C., Neto, J.C. & Melo, Í.D.L. Fenchel Conjugate via Busemann Function on Hadamard Manifolds. Appl Math Optim 88, 83 (2023). https://doi.org/10.1007/s00245-023-10060-y

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