We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for Hörmander vector fields.

A simple arc gamma subset of R-n is called a Whitney arc if there exists a non-constant real function f on gamma such that lim(y -> x),(y is an element of gamma) vertical bar f(y) - f(x)vertical bar/vertical bar y - x vertical bar = 0 for every x is an element of gamma; gamma is 1-critical if there exists an f is an element of C-1(R-n) such that f'(x) = 0 for every x is an element of gamma and f is not constant on gamma. We show that the two notions are equivalent if gamma is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc gamma in R-2 each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney arc. This answers completely a problem of G. Petruska which was solved for n >= 3 by the first author in 1999.

Using results and methods of G. Choquet (1944