  In this paper, we endow the space of continuous translation invariant
valuation on convex sets generated by mixed volumes coupled with a suitable
Radon measure on tuples of convex bodies with two appropriate norms. This
enables us to construct a continuous extension of the convolution operator on
smooth valuations to non-smooth valuations, which are in the completion of the
spaces of valuations with respect to these norms.
  The novelty of our approach lies in the fact that our proof does not rely on
the general theory of wave fronts, but on geometric inequalities deduced from
optimal transport methods.
  We apply this result to prove a variant of Minkowski's existence theorem, and
generalize a theorem of Favre-Wulcan and Lin in complex dynamics over toric
varieties by studying the linear actions on the Banach spaces of valuations and
by studying their corresponding eigenspaces.
