Valuations on Sobolev spaces


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.