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Valuations on Sobolev spaces

From: American Journal of Mathematics
Volume 134, Number 3, June 2012
pp. 827-842 | 10.1353/ajm.2012.0019

Abstract

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.