On extendability by continuity of valuations on convex polytopes

There is a well known construction of weakly continuous valuations on convex compact polytopes in Rn. In this paper we investigate when a special case of this construction gives a valuation which extends by continuity in the Hausdorff metric to all convex compact subsets of Rn. It is shown that there is a necessary condition on the initial data for such an extension. In the case of R3 more explicit results are obtained.