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In this paper we consider the $\chi$-function (the Holevo capacity ofconstrained channel) and the convex closure of the output entropy for arbitraryinfinite dimensional channel.  It is shown that the $\chi$-function of an arbitrary channel is a concavelower semicontinuous function on the whole state space, having continuousrestriction to any set of continuity of the output entropy.  The explicit representation for the convex closure of the output entropy isobtained and its properties are explored. It is shown that the convex closureof the output entropy coincides with the convex hull of the output entropy onthe convex set of states with finite output entropy. Similarly to the case ofthe $\chi$-function, it is proved that the convex closure of the output entropyhas continuous restriction to any set of continuity of output entropy. Someapplications of these results to the theory of entanglement are discussed.  The obtained properties of the convex closure of the output entropy make itpossible to generalize to the infinite dimensional case the convex dualityapproach to the additivity problem.Comment: 25 page

On entropic quantities related to the classical capacity of infinite dimensional quantum channels