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.