Positivity of valuations on convex bodies and invariant valuations by linear actions

38 pages; v2: references updated, typos corrected, Remark 1.2 and Section 6.2.1 addedWe define a notion of positivity on continuous and translation invariant valuations on convex bodies on a finite dimensional real vector space. We endow the valuation space generated by mixed volumes with a norm induced by the positive cone. This enables us to construct a continuous extension of the convolution operator on smooth valuations to the closure of that space. As an application, we prove a variant of Minkowski's existence theorem. Furthermore, given a linear map, we generalize a theorem of Favre-Wulcan and Lin by proving that the eigenvalues of the linear map is related to the spectral radius of the induced linear operator on the space of valuations. Finally, given a linear action and under a natural strict log-concavity assumption on certain spectral radius of the induced linear operators on valuations, we study the positivity properties of the space of invariant valuations corresponding to the spectral radius of the operator