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Examples of convex functions and classifications of normed spaces. (English) Zbl 0823.46005

Summary: We study various properties of convex functions and their connections to the structure of the spaces on which they are defined. In particular, it is shown boundedness properties of convex functions on various bornologies are related to sequential convergence in dual topologies. Convex functions whose subdifferentials have range with nonconvex interior are constructed on nonreflexive spaces, and we exhibit examples of convex functions on infinite-dimensional spaces whose subdifferentials have sparse domains.

MSC:

46A17 Bornologies and related structures; Mackey convergence, etc.
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
49J52 Nonsmooth analysis
46B20 Geometry and structure of normed linear spaces
52A41 Convex functions and convex programs in convex geometry
46G05 Derivatives of functions in infinite-dimensional spaces
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