Automatic convexity

Abstract

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.