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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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 827-842
Launched on MUSE
2012-05-31
Open Access
No
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