All affinely covariant convex-body-valued valuations on the Sobolev space W1,1(Rⁿ) 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∈ W1,1(Rⁿ) the unit ball of its optimal Sobolev norm.
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Monika Ludwig (2012) studied this question.
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