Topology of singular set of semiconcave function via Arnaud's theorem

We proved the (local) path-connectedness of certain subset of the singular set of semiconcave functions with linear modulus in general. In some sense this result is optimal. The proof is based on a theorem by Marie-Claude Arnaud (M.-C. Arnaud, $\mathit{\text{Pseudographs}}\mathit{\text{and}}\mathit{\text{the}}\mathit{\text{Lax-Oleinik}}\mathit{\text{semi-group:}}\mathit{\text{a}}\mathit{\text{geometric}}\mathit{\text{and}}\mathit{\text{dynamical}}\mathit{\text{interpretation}}$. Nonlinearity, $\mathbf{\text{24}}$(1): 71-78, 2011.). We also gave a new proof of the theorem in time-dependent case.