The affine Sobolev–Zhang inequality on BV(Rn)BV(Rn)

The affine Sobolev–Zhang inequality is extended to BV(Rn)BV(Rn), the space of functions of bounded variation on RnRn, and the equality cases are characterized. As a consequence, the Petty projection inequality for sets of finite perimeter, which implies the isoperimetric inequality for sets of finite perimeter, is established.