Moser-Trudinger inequalities without boundary conditions and isoperimetric problems

The best constant is exhibited in Trudinger’s exponential inequality for functions from the Sobolev space W^{1,n}(G),where G is an open subset of R^n, and n >1. This complements a classical result by Moser dealing with the subspace W^{1,n}_0(G). An extension to the borderline Lorentz-Sobolev spaces W^1L^{n,q}(G), with q between 1 and infinity is also established. A key step in our proofs is an asymptotically sharp relative isoperimetric inequality for domains in R^n.