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Journal of Convex Analysis 10 (2003), No. 1, 275--284
Copyright Heldermann Verlag 2003

Automatic Convexity

Charles A. Akemann
Dept. of Mathematics, University of California, Santa Barbara, CA 93106, U.S.A.,

Nik Weaver
Dept. of Mathematics, Washington University, St. Louis, MO 63130, U.S.A.,

In many cases the convexity of the image of a linear map with range is Rn 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 the intersection of K and H have the form of intersections of F and H for any 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.

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