锥的方向精细族：构造原理、逼近与分离
Directionally Fine Families of Cones: Construction Principles, Approximation, and Separation
查看机构详情
- University of Alicante（阿利坎特大学）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
本文提出锥的方向精细族概念，作为有限分离与逼近的统一机制，通过Bishop--Phelps锥和横向强制原理实现，并应用于局部紧锥优化的惩罚与恢复。
中文摘要 AI 辅助
我们引入了锥的方向精细族的概念，这是一种局部几何性质，表达了在赋范空间的每个方向周围选择任意窄锥的可能性。我们证明了这一局部性质为有限分离和逼近提供了共同机制。特别地，方向紧集可以通过该族的有限多个成员与闭锥分离，而局部紧锥则允许有限外部逼近，并对其方向集和有界截面上具有定量的Hausdorff控制。这些有限构造自然导致极大型正齐次分离器，且不要求被分离的锥具有凸性。然后我们研究了该框架的两种互补实现。对于经典的Bishop--Phelps锥族，方向精细性由单位球面上的每一点都是单位球的齿点这一要求来刻画，等价于性质$(G)$；在Banach空间中，这相当于轮转性连同Kadec性质。另一方面，我们引入了横向强制作为一般的轴向构造原理。它在任意赋范空间中产生方向精细族，无需对范数作几何假设，并将均匀轴向偏差和范数归一化轴向锥作为典型模型。最后，前者的显式结构为局部紧锥上的优化提供了惩罚和恢复结果，以及定量逼近和收敛估计。
英文摘要
We introduce the notion of a directionally fine family of cones, a local geometric property expressing the possibility of selecting arbitrarily narrow cones around every direction of a normed space. We show that this local property provides a common mechanism for finite separation and approximation. In particular, directionally compact sets can be separated from closed cones by finitely many members of the family, while locally compact cones admit finite outer approximations with quantitative Hausdorff control on their sets of directions and on bounded sections. These finite constructions lead naturally to max-type positively homogeneous separators and do not require convexity of the cones to be separated. We then study two complementary realizations of the framework. For the classical family of Bishop--Phelps cones, directional fineness is characterized by the requirement that every point of the unit sphere be a denting point of the unit ball, equivalently, by property~$(G)$; in Banach spaces this amounts to rotundity together with the Kadec property. On the other hand, we introduce transversal coercivity as a general axial construction principle. It yields directionally fine families in arbitrary normed spaces, without geometric assumptions on the norm, and includes uniform axial deviation and norm-normalized axial cones as canonical models. Finally, the explicit structure of the former yields penalization and recovery results for optimization over locally compact cones, together with quantitative approximation and convergence estimates.
