  We consider the class of smooth convex functions defined over an open convex
set. We show that this class is essentially different than the class of smooth
convex functions defined over the entire linear space by exhibiting a function
that belongs to the former class but cannot be extended to the entire linear
space while keeping its properties. We proceed by deriving new properties of
the class under consideration, including an inequality that is strictly
stronger than the classical Descent Lemma.
