Continuous valuations on the space of Lipschitz functions on the sphere

We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere Sn−1. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with...

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Bibliographic Details
Published in:Journal of functional analysis 2021-02, Vol.280 (4), p.108873, Article 108873
Main Authors: Colesanti, Andrea, Pagnini, Daniele, Tradacete, Pedro, Villanueva, Ignacio
Format: Article
Language:English
Subjects:
Geometric valuation theory
Integral representation
Lipschitz functions
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Summary:We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere Sn−1. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with measure theoretical arguments, will yield an integral representation for continuous and rotation invariant valuations on the space of Lipschitz functions over the 1-dimensional sphere.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2020.108873