  It is shown that every even, zonal measure on the Euclidean unit sphere gives
rise to an isoperimetric inequality for sets of finite perimeter which directly
implies the classical Euclidean isoperimetric inequality. The strongest member
of this large family of inequalities is shown to be the only affine invariant
one among them - the Petty projection inequality. As an application, a family
of sharp Sobolev inequalities for functions of bounded variation is obtained,
each of which is stronger than the classical Sobolev inequality. Moreover,
corresponding families of Lp isoperimetric and Sobolev type inequalities are
also established.
