Nearly convex sets: fine properties and domains or ranges of subdifferentials of convex functions

TL;DR

This paper systematically studies nearly convex sets, highlighting their importance in convex analysis and providing examples of subdifferentials with nonconvex domains or ranges, advancing understanding of their properties.

Contribution

It offers a comprehensive analysis of nearly convex sets and constructs examples of subdifferentials with nonconvex domains or ranges, which was previously less understood.

Findings

Nearly convex sets have significant roles in convex analysis.

Examples of subdifferentials with nonconvex domains/ranges are constructed.

The paper advances the theoretical understanding of nearly convex sets.

Abstract

Nearly convex sets play important roles in convex analysis, optimization and theory of monotone operators. We give a systematic study of nearly convex sets, and construct examples of subdifferentials of lower semicontinuous convex functions whose domain or ranges are nonconvex.