Dot product invariant valuations on Lip$(S^{n-1})$

TL;DR

This paper develops an integral representation for a class of valuations on Lipschitz functions on the sphere that are invariant under rotations and dot products, advancing the understanding of geometric valuations.

Contribution

It introduces a novel integral representation for continuous, rotation and dot product invariant valuations on Lip$(S^{n-1})$, expanding the theoretical framework of valuation theory.

Findings

Provides a new integral representation for these valuations.

Characterizes the invariance properties under rotation and dot product.

Enhances the mathematical understanding of valuations on Lipschitz functions.

Abstract

We provide an integral representation for continuous, rotation invariant and dot product invariant valuations defined on the space Lip$(S^{n-1})$ of Lipschitz continuous functions on the unit $n-$sphere.