Sharp Sobolev Inequalities via Projection Averages.

A family of sharp L p Sobolev inequalities is established by averaging the length of i-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical L p  Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them-the affine L p  Sobolev inequality of Lutwak, Yang, and Zhang. When p = 1 , the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extremal functions in this case.