  In this paper we consider the $\chi$-function (the Holevo capacity of
constrained channel) and the convex closure of the output entropy for arbitrary
infinite dimensional channel.
  It is shown that the $\chi$-function of an arbitrary channel is a concave
lower semicontinuous function on the whole state space, having continuous
restriction to any set of continuity of the output entropy.
  The explicit representation for the convex closure of the output entropy is
obtained and its properties are explored. It is shown that the convex closure
of the output entropy coincides with the convex hull of the output entropy on
the convex set of states with finite output entropy. Similarly to the case of
the $\chi$-function, it is proved that the convex closure of the output entropy
has continuous restriction to any set of continuity of output entropy. Some
applications of these results to the theory of entanglement are discussed.
  The obtained properties of the convex closure of the output entropy make it
possible to generalize to the infinite dimensional case the convex duality
approach to the additivity problem.
