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 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.

在本文中，我们赋予了由混合体积生成的凸集的连续平移不变估值的空间，并结合了具有两个适当范数的凸体元组上的适当Radon度量。这使我们能够构造从平滑估值到非平滑估值的卷积运算符的连续扩展，这是相对于这些规范的估值空间的完成。我们的方法的新颖性在于，我们的证明不依赖于波前的一般理论，而是依赖于由最佳传输方法得出的几何不等式。我们用这个结果来证明Minkowski存在定理的一个变体，