Applications of Convex Analysis within Mathematics

数学公案与嘉当凸性：$\Gamma$-凸包与蝶形实现

Mathematical Koans and Cartan Convexity: $Γ$-Convex Hulls and Butterfly Realizations

本文致敬侯世达定义数学公案，针对自伴自由函数引入嘉当凸性，结合非交换Kraus-蝶形定理推导相关凸性与蝶形实现结论，区分提升条件与内在$\Gamma$-凸性。

局部凸空间中的拉格朗日乘子

Lagrange Multipliers in locally convex spaces

本文研究了Hausdorff局部凸空间中含无穷多约束的数学规划问题，提出一般拉格朗日乘子规则，用Gateaux可微性替代Fréchet可微性，并引入“容许集”替代闭凸锥以推广现有结果。

A structural characterization of numeraires of convex sets of nonnegative random variables

We introduce the concept of numeraires of convex sets in the nonnegative orthant of the topological vector space of all random variables built over a probability space. A necessary and sufficient condition for an element of a convex set to be its numeraire is given, inspired from ideas in financial mathematics.

Construction of bipotentials and a minimax theorem of Fan

In Mechanics, the theory of standard materials is a well-known application of Convex Analysis. However, the so-called non-associated constitutive laws cannot be cast in the mould of the standard materials. From the mathematical viewpoint, a non associated constitutive law is a multivalued operator which is not supposed to be monotone. A possible way to study non-associated constitutive laws by using Convex Analysis, proposed first in [12], consists in constructing a "bipotential" function of two variables, which physically represents the dissipation. This is a second paper on the mathematics o…

实拓扑线性空间中的非线性锥分离定理

Nonlinear Cone Separation Theorems in Real Topological Linear Spaces

本文基于增广对偶锥和Bishop-Phelps型分离函数，在实拓扑线性空间中推导出针对两个非凸锥的非线性锥分离新定理，其证明以凸集经典分离定理为关键工具。

Weak$^{\ast }$ Hypertopologies with Application to Genericity of Convex Sets

We propose a new class of hypertopologies, called here weak$^{\ast }$ hypertopologies, on the dual space $\mathcal{X}^{\ast }$ of a real or complex topological vector space $\mathcal{X}$. The most well-studied and well-known hypertopology is the one associated with the Hausdorff metric for closed sets in a complete metric space. Therefore, we study in detail its corresponding weak$^{\ast }$ hypertopology, constructed from the Hausdorff distance on the field (i.e. $\mathbb{R}$ or $\mathbb{C}$) of the vector space $\mathcal{X}$ and named here the weak$^{\ast }$-Hausdorff hypertopology. It has no…