Valuations on Sobolev Spaces

All affinely covariant convex-body-valued valuations on the Sobolev space W (R) 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 ∈W (R) the unit ball of its optimal Sobolev norm. 2000 AMS subject classification: 46B20 (46E35, 52A21,52B45) Let ‖ ·‖ denote a norm on Rn that is normalized so that its unit ball has the same volume, vn, as the n-dimensional Euclidean unit ball. For such a norm, the sharp Gagliardo-NirenbergSobolev inequality states that ∫ Rn ‖∇f(x)‖∗ dx ≥ n v n |f | n n−1 (1) for every f ∈ W 1,1(Rn). Here for p ≥ 1, |f |p denotes the Lp norm of f and ‖ · ‖∗ the dual norm of ‖ · ‖ (see Section 1 for precise definitions). The Sobolev space W 1,1(Rn) is the space of functions f ∈ L1(Rn) such that their weak gradient ∇f is in L1(Rn). If the unit ball B of ‖ · ‖ is the Euclidean unit ball, then inequality (1) goes back to Federer and Fleming [15] and Maz′ya [46] and is known to be equivalent to the Euclidean isoperimetric inequality. For general norms, (1) was established by Gromov [49, Appendix]. Note that the right hand side of (1) does not depend on ‖ · ‖. Hence for a given f ∈ W 1,1(Rn), n ≥ 2, we may ask for its optimal Sobolev norm, that is, for the norm that minimizes the left-hand side of (1) among all norms whose unit balls have volume vn. This natural and important question was first asked by Lutwak, Yang and Zhang in [45]. They showed that the unit ball 〈f〉 corresponding to the optimal Sobolev norm of f ∈ W 1,1(Rn) is (up to normalization) the unique origin-symmetric convex body (that is, compact, convex set) in Rn such that ∫ Sn−1 g(u) dS(〈f〉, u) = ∫