Colesanti, A., Knörr, J., & Pagnini, D. (2024). The homogeneous decomposition of dually translation invariant valuations on Lipschitz functions on the sphere. arXiv. https://doi.org/10.48550/arXiv.2401.05913
We show that every continuous and dually translation invariant valuation on the space of Lipschitz functions on the unit sphere of $\mathbb{R}^n$, $n\ge2$, can be decomposed uniquely into a sum of homogeneous valuations of degree $0$, $1$ and $2$. In particular, there does not exist any non-trivial, continuous and dually translation invariant valuation which is homogeneous of degree $3$ or higher. For the space of those of degree $0$, $1$ and $2$ we provide a description of a dense subspace.
