On Quermassintegrals of mixed projection bodies

One of the major outstanding questions in Geometric Convexity is Petty's conjectured inequality between the volume of a convex body and that of its projection body. It is shown that if Petty's conjectured inequality holds, then it is the first of a family of such inequalities (involving mixed projection bodies). All of the members of this family are strengthened versions of the classical inequalities between pairs of Quermassintegrals of a convex body. The last member of this family (of conjectured inequalities) is established.