ar X iv : m at h / 01 05 15 6 v 1 [ m at h . FA ] 1 7 M ay 2 00 1 Automatic Convexity

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 ∩ H have the form F ∩ 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. Historically there have been several theorems that concluded, unexpectedly, even mysteriously at first, that a certain set in R n is convex. Perhaps the two best known examples are the convexity of the numerical range of an n × n complex matrix [Hau, T] and Lya-punov's theorem on the convexity of the range of a vector measure [Ly]. In each of these cases the set in question is the image under some apparently non-linear map of a non-convex set. Each of these theorems has been generalized in many directions. Until the work of Lindenstrauss [Li], Lyapunov's theorem remained a mystery with several complicated , yet incomplete, proofs (including Lyapunov's and a later proof by Halmos [Hal-1]) in the literature. See [AA] for a discussion of Lyapunov's theorem and generalizations. As for the convexity of the numerical range, while the proofs in the literature have been complete , and they have gotten steadily simpler, the mystery of the appearance of convexity has remained (see [HJ, p. 78], [P] and [GR, sections 1.1 and 5.5]). In [AA] a number of automatic convexity …