Positivity of Valuations on Convex Bodies and Invariant Valuations by Linear Actions

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