  Level proximal subdifferential was introduced by Rockafellar recently as a
tool for studying proximal mappings of possibly nonconvex functions. In this
paper we give a systematic study of level proximal subdifferntial, characterize
variational convexity of the function by locally firm nonexpansiveness of
proximal mappings or locally relative monotonicity of level proximal
subdifferential, and investigate pointwise Lipschitz smoothness of the
function. Integration and single-valuedness of level proximal subdifferential
are also examined.
