从序到凸性：面理论与广义分离
Convexity from order: a theory of faces and generalized separation
- University of British Columbia（不列颠哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文从序理论视角研究凸集，将面视为强最小值集合，通过定向映射实现广义分离，并定义联合支撑子空间，在仅假设凸性下推广了面运算结果，解决了开放问题。
AI中文摘要：
已知一般向量空间中的任何凸锥都是与该向量空间相容的预序的正锥，并且任何凸集都是凸锥的仿射切片。因此，我们将凸集视为正锥的仿射切片，从序理论的角度处理凸集。我们发现面是强最小值的集合，并将泛函推广到取值于任意有序向量空间的定向映射。定向映射暴露每一个面，并且当 $f(D) \le 0 \le f(C)$ 逐元素成立时，定向仿射映射 $f$ 将 $C$ 与 $D$ 分离。这种超平面分离的推广恢复了“局部”的数学对象，包括点处生成的面和法锥。分离定向映射的核构成一个分离仿射平坦格的格，其底元素是联合支撑子空间（JSS），该概念最近在有限维中引入，用于无资格条件的凸分析结果。我们在一般向量空间中定义两个凸集 $C, D$ 的 JSS 为交叉线性性 $\mathrm{lin}(K-G)$ 的仿射切片，其中 $K, G$ 分别是 $C, D$ 的齐次化锥。我们证明在有限维中，双边面约简对应于分离仿射平坦格中的下降，我们将其形式化为定向映射的字典序积。面的字典序刻画作为推论随之而来。通过仅假设凸性，我们的方法产生了一般性结果：我们证明两个凸集之差的凸集的面是两个面的差，去掉了紧性、有限维性和可暴露性的假设。我们还证明两个凸集交集的任何面是两个面的交集，解决了 Weis 在“关于凸集的面的一则注记”中提出的开放问题 5.10，去掉了非空内在核的假设。
英文摘要：
It is known that any convex cone in a general vector space is the positive cone of a preorder compatible with the vector space, and that any convex set is an affine slice of a convex cone. We therefore treat convex sets order-theoretically, as affine slices of positive cones. We find that faces are sets of strong minima, and we generalize functionals to oriented maps valued in arbitrary ordered vector spaces. Oriented maps expose every face, and an oriented affine map $f$ separates $C$ from $D$ when $f(D) \le 0 \le f(C)$ element-wise. This generalization of hyperplane separation recovers mathematical objects that are "local", including the generated face at a point and the normal cone. The kernels of separating oriented maps form a lattice of separating affine flats whose bottom element is the joint supporting subspace (JSS), recently introduced in finite dimensions for qualification-free convex analysis results. We define the JSS in general vector spaces for two convex sets $C, D$ as the affine slice of the cross-lineality $\mathrm{lin}(K-G)$, where $K,G$ are the homogenization cones of $C, D$. We show that in finite dimensions, bilateral facial reduction corresponds to a descent in the lattice of separating affine flats, which we formalize as lexicographic products of oriented maps. The lexicographic characterization of faces follows as a corollary. By making no assumption other than convexity, our approach yields general results: we show that the face of a difference of two convex sets is a difference of faces, dropping assumptions of compactness, finite dimensionality, and exposedness. We also show that any face of an intersection of two convex sets is an intersection of faces, resolving open problem 5.10 by Weis in "A note on faces of convex sets", dropping the assumption of non-empty intrinsic core.
