Optimal Sobolev norms and the Lp Minkowski problem

The existence and uniqueness of an optimal Lp Sobolev norm for a function on ℝn is shown to be essentially equivalent to the existence and uniqueness of the solution to the Lp Minkowski problem for even measures. The former is established using the latter. This leads to new affine analytic inequalities as well as a new proof of the affine Lp Sobolev inequality previously established by the authors.