  In the first part of the note we prove that a sufficient condition (due to
Simons) for the convexity of the closure of the domain/range of a monotone
operator is also necessary when the operator has bounded domain and is maximal.
Simons' condition is closely related to the notion of regular maximal monotone
operator. In the second part of the note we give several characterizations for
the regularity of a maximal monotone operator, show that a maximal monotone
operator of type (FPV) is regular and improve a previous sum theorem type
result.
