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.

在本文中，我们引入了 Fenchel 型共轭，通过 Busemann 函数作为凸函数的上界给出。众所周知，布斯曼函数是具有恒定范数梯度的光滑凸函数。我们的研究确保我们关于 Fenchel 共轭的建议最充分地解决了 Hadamard 流形上非恒定仿射函数近似的缺失。更准确地说，作为本文的第一个贡献，我们证明任何仿射函数在某些开集上的非正里奇曲率的完整连通黎曼流形中都是常数。此外，我们还展示了截面曲率对获得主要结果的影响。特别地，我们说明了真函数 lsc 凸函数与其双共轭函数之间的差异是一个常数，该常数取决于流形的截面曲率，这表明 Fenchel-Moreau 型定理通常直接受截面曲率的影响。我们还介绍了一些根据 Fenchel 结合物配制的应用。