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Valuations on Sobolev spaces
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 134, Number 3, June 2012
- pp. 827-842
- 10.1353/ajm.2012.0019
- Article
- Additional Information
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Abstract
All affinely covariant convex-body-valued valuations on the Sobolev space
$W^{1,1}({\Bbb{R}}^n)$ are completely classified. It is shown that there
is a unique such valuation for Blaschke addition. This valuation turns out
to be the operator which associates with each function $f\in
W^{1,1}({\Bbb{R}}^n)$ the unit ball of its optimal Sobolev norm.
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