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arXiv · 2401.05913

The homogeneous decomposition of dually translation invariant valuations on Lipschitz functions on the sphere

Andrea Colesanti·Jonas Knoerr·Daniele Pagnini
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Abstract

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.

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Andrea Colesanti, Jonas Knoerr, Daniele Pagnini. 2024-01-11. The homogeneous decomposition of dually translation invariant valuations on Lipschitz functions on the sphere. https://arxiv.org/abs/2401.05913

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