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We consider two properties of the set of quantum states as a convextopological space and some their implications concerning the notions of aconvex hull and of a convex roof of a function defined on a subset of quantumstates.  By using these results we analyze two infinite-dimensional versions (discreteand continuous) of the convex roof construction of entanglement monotones,which is widely used in finite dimensions. It is shown that the discreteversion may be 'false' in the sense that the resulting functions may notpossess the main property of entanglement monotones while the continuousversion can be considered as a 'true' generalized convex roof construction. Wegive several examples of entanglement monotones produced by this construction.In particular, we consider an infinite-dimensional generalization of the notionof Entanglement of Formation and study its properties.

On properties of the space of quantum states and their application to construction of entanglement monotones