Radial continuous valuations on star bodies

We show that a radial continuous valuation defined on the n-dimensional star bodies extends uniquely to a continuous valuation on the n-dimensional bounded star sets. Moreover, we provide an integral representation of every such valuation, in terms of the radial function, which is valid on the dense...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2017-10, Vol.454 (2), p.995-1018
Main Authors: Tradacete, Pedro, Villanueva, Ignacio
Format: Article
Language:English
Subjects:
Convex geometry
Star bodies
Valuations
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Summary:We show that a radial continuous valuation defined on the n-dimensional star bodies extends uniquely to a continuous valuation on the n-dimensional bounded star sets. Moreover, we provide an integral representation of every such valuation, in terms of the radial function, which is valid on the dense subset of the simple Borel star sets. Along the way, we also show that every radial continuous valuation defined on the n-dimensional star bodies can be decomposed as a sum V=V+−V−, where both V+ and V− are positive radial continuous valuations.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2017.05.026