Automatic convexity

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dc.creatorAkemann, Charles A.
dc.creatorWeaver, Nik
dc.date2001-05-17
dc.date.accessioned2026-07-07T04:41:46Z
dc.date.available2026-07-07T04:41:46Z
dc.descriptionIn 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.
dc.description10 pages
dc.identifierhttps://arxiv.org/abs/math/0105156
dc.identifierhttp://arxiv.org/abs/math/0105156
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/61495
dc.subjectFunctional Analysis
dc.subjectOperator Algebras
dc.subject46A55; 47A12; 46G10
dc.titleAutomatic convexity
dc.typetext

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