Nearly convex sets: fine properties and domains or ranges of subdifferentials of convex functions. (English) Zbl 1355.52005
The paper provides a systematic study of nearly convex sets (also known in the literature as almost convex sets), a class of sets that generalize the one of the convex sets and has applications in convex analysis, optimization and theory of monotone operators, among others. After presenting basic notations and facts about convex sets and nearly convex sets in Section 2, the authors give in Section 3 new characterizations of nearly convex sets. In Section 4, one can find calculus of nearly convex sets and relative interiors, having as a byproduct a new proof of the maximality of a sum of several maximally monotone operators. Recession sets of nearly convex sets and closedness of nearly convex sets under a linear mapping are studied in Section 5, while in the next one examples of proper lower semicontinuous convex functions having nearly convex sets as subdifferential domains are constructed. Furthermore, some open problems are presented in Section 7, while an Appendix contains detailed proofs of some results from Section 6.
Reviewer: Sorin-Mihai Grad (Chemnitz)
MSC:
| 52A41 | Convex functions and convex programs in convex geometry |
| 26A51 | Convexity of real functions in one variable, generalizations |
| 47H05 | Monotone operators and generalizations |
| 47H04 | Set-valued operators |
| 52A30 | Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.) |
Keywords:
maximally monotone operators; nearly convex sets; nearly equal sets; relative interior; recession cones; subdifferentials; almost convex setsSoftware:
GeoGebraReferences:
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