The dimension of semialgebraic subdifferential graphs

TL;DR

This paper investigates the local dimensionality of subdifferential graphs of semi-algebraic functions, revealing they have constant local dimension, contrasting with general functions, and explores exceptions for Clarke subdifferentials.

Contribution

It establishes that semi-algebraic functions have subdifferential graphs with constant local dimension, extending understanding of subdifferential structure in semi-algebraic analysis.

Findings

Semi-algebraic functions have subdifferential graphs with constant local dimension n.

Examples show that general functions can have large, complex subdifferential graphs.

The result relates to Minty's theorem and highlights differences for Clarke subdifferentials.

Abstract

Examples exist of extended-real-valued closed functions on ${\bf R}^n$ whose subdifferentials (in the standard, limiting sense) have large graphs. By contrast, if such a function is semi-algebraic, then its subdifferential graph must have everywhere constant local dimension $n$. This result is related to a celebrated theorem of Minty, and surprisingly may fail for the Clarke subdifferential.