子微分理论与通过Busemann函数的Fenchel共轭在Hadamard流形上
Busemann-coupling subdifferentials and Fenchel biconjugation on Hadamard manifolds
AI总结：
本文通过Busemann函数定义子微分，研究Fenchel-Young不等式成立条件及非平凡仿射函数的存在性，揭示了凸分析中的对偶原理和里奇曲率对刚性结论的必要性。
摘要：
在本文中，我们提出了一种通过Busemann函数定义的子微分概念，并利用该概念确定Bento、Cruz Neto和Melo（Appl. Math. Optim. 88:83，2023）提出的Fenchel-Young不等式在何种条件下成立等号。这一等号条件尤其重要，因为它捕捉了凸分析中的一个基本对偶原理，将一个原凸函数与其共轭函数联系起来，并阐明了在Riemann流形上相关不等式的精确性。我们还研究了在里奇曲率信息下非平凡仿射函数的存在性。特别是，我们将Bento、Cruz Neto和Melo最初在开放集上负里奇曲率情况下的结果扩展到里奇曲率可能非零的流形上。作为结果，我们证明了非平凡仿射函数的新不存在准则，并表明非零里奇曲率的假设通常对于确保此类刚性结论是必要的。
英文摘要：
We introduce an intrinsic subdifferential on Hadamard manifolds induced by a base-point-dependent Busemann coupling, motivated by the scarcity of nonconstant affine functions on non-Euclidean Hadamard manifolds. For proper functions, it characterizes exactly the primal-dual equality pairs in the Fenchel-Young inequality of Bento, Cruz Neto, and Melo (\emph{Appl. Math. Optim.} 88:83, 2023). A dual-point--tangent-vector bijection yields Fermat's rule and isometry covariance. We characterize everywhere nonemptiness for finite-valued functions by pointwise-attained \(H_p\)-envelope representations, encompassing the radial models with explicit subdifferentials and the nonsmooth distance from the base point. In hyperbolic space, base-point rigidity shows that, for functions differentiable at the base point, nonemptiness is equivalent to global minimality and vanishing of the corresponding Fenchel-Young gap. Two geodesically convex functions on the Poincaré disk prove strict noninclusions between our construction and fixed-direction Busemann subdifferentials, although both recover the classical Euclidean subdifferential. This separation is explained by coupling orientation. In constant negative curvature, a strictly increasing scalar profile describes the asymmetry and defines an intrinsic measure of nonlinearity. We compute this measure at arbitrary radius and derive a lower bound and two-sided estimates for the biconjugation gap; at unit radius, the exact curvature-dependent bound is attained by deviations of both signs.
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