  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.
