The dimension of semialgebraic subdifferential graphs
  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. 
Dmitriy Drusvyatskiy, Alexander D. Ioffe, Adrian S. Lewis
math.OC math.AG