A convex, finite and lower semicontinuous function with empty subdifferential

TL;DR

This paper presents a counterexample of a convex, lower semicontinuous function with an empty subdifferential everywhere, highlighting the importance of completeness in convex analysis.

Contribution

It introduces a novel example of such a function defined on an incomplete space, challenging assumptions in convex analysis.

Findings

Counterexample exists in incomplete normed spaces

Subdifferential can be empty everywhere for certain convex functions

Completeness is essential for some convex analysis results

Abstract

We give an example of a convex, finite and lower semicontinuous function whose subdifferential is everywhere empty. This is possible since the function is defined on an incomplete normed space. The function serves as a universal counterexample to various statements in convex analysis in which completeness is required.