自动凸度
Automatic convexity
摘要：
在许多情况下，具有范围为$R^n$的线性映射的图像的凸性是自动的，因为映射的域的面部结构。我们开发了一个四步程序来证明这种“自动凸性”。为了使这一过程更有效，我们证明了两个新的定理，确定的面部结构的交集的凸集与子空间的原始集的面部结构。令$K$是实线性空间$X$中的凸集，并且令$H$是满足$K$的X的子空间。在第一部分中，我们证明了$K\cap H$的面具有$F\cap H$的形式，对于$K$的面$F$。然后将交定理推广到局部凸线性拓扑空间$X$，$K$和$H$是闭的，且$H$在$X$中有有限余维数的情形.在第二部分中，我们使用我们的程序来“解释”一个复矩阵的数值范围（及其一些推广）的凸性。在第三部分中，我们使用我们的相交定理的拓扑版本来证明一个版本的李雅普诺夫定理与众多的线性约束。我们还将Samet的连续提升定理推广到同样的约束情形。
英文摘要：
In many cases the convexity of the image of a linear map with range is $R^n$ is automatic because of the facial structure of the domain of the map. We develop a four step procedure for proving this kind of ``automatic convexity''. To make this procedure more efficient, we prove two new theorems that identify the facial structure of the intersection of a convex set with a subspace in terms of the facial structure of the original set. Let $K$ be a convex set in a real linear space $X$ and let $H$ be a subspace of X that meets $K$. In Part I we show that the faces of $K\cap H$ have the form $F\cap H$ for a face $F$ of $K$. Then we extend our intersection theorem to the case where $X$ is a locally convex linear topological space, $K$ and $H$ are closed, and $H$ has finite codimension in $X$. In Part II we use our procedure to ``explain'' the convexity of the numerical range (and some of its generalizations) of a complex matrix. In Part III we use the topological version of our intersection theorem to prove a version of Lyapunov's theorem with finitely many linear constraints. We also extend Samet's continuous lifting theorem to the same constrained siuation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
