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Journal of Convex Analysis 19 (2012), No. 2, 467--483
Copyright Heldermann Verlag 2012

Remarks on the Γ--regularization of Non-convex and Non-semi-continuous Functions on Topological Vector Spaces

Jean-Bernard Bru
Dep. de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain

Walter de Siqueira Pedra
Institut für Mathematik, Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany

We show that the minimization problem of any non-convex and non-lower semi-continuous function on a compact convex subset of a locally convex real topological vector space can be studied via an associated convex and lower semi-continuous function Γ(h). This observation uses the notion of Γ-regularization as a key ingredient. As an application we obtain, on any locally convex real space, a generalization of the Lanford III--Robinson theorem which has only been proven for separable real Banach spaces. The latter is a characterization of subdifferentials of convex continuous functions.

Keywords: Variational problems, non-linear analysis, non-convexity, Gamma-regularization, Lanford III -- Robinson theorem.

MSC: 58E30, 46N10, 52A07

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