The study of affine isoperimetric inequalities in con- vex geometry has lead to deeper understanding of several Sobolev type inequalities. As a completion of the work in this direction, several related analytic and geometric questions are discussed.<br />In Chapter 1, The affine Sobolev-Zhang inequality is extended to BV(R n), the space of functions of bounded variation on Rn, and the equality cases are characterized. As a consequence, the Petty projection inequality for sets of ¯nite perimeter, which implies the isoperimetric inequality for sets of ¯nite perimeter, is established.<br />In Chpater 2, All affinely covariant convex-body-valued semi- valuations on functions of bounded variation on Rn are completely classiffied. It is shown that there is a unique such semi-valuation for Blaschke addition. This semi-valuation turns out to be the operator which associates with each function its extended LYZ body.<br />In Chaper 3, A Brothers-Ziemer type theorem for the a±ne Polya-Szego principle and a quantitative affine Polya-Szego prin- ciple are established.
