  The first part of the paper provides new characterizations of the normal cone
to the effective domain of the supremum of an arbitrary family of convex
functions. These results are applied in the second part to give new formulas
for the subdifferential of the supremum function, which use both the active and
nonactive functions at the reference point. Only the data functions are
involved in these characterizations, the active ones from one side, together
with the nonactive functions multiplied by some appropriate parameters. In
contrast with previous works in the literature, the main feature of our
subdifferential characterization is that the normal cone to the effective
domain of the supremum (or to finite-dimensional sections of this domain) does
not appear. A new type of optimality conditions for convex optimization is
established at the end of the paper.
