Mixed projection inequalities

Abstract

A number of sharp geometric inequalities for polars of mixed projection bodies (zonoids) are obtained. Among the inequalities derived is a polar projection inequality that has the projection inequality of Petty as a special case. Other special cases of this polar projection inequality are inequalities (between the volume of a convex body and that of the polar of its i i th projection body) that are strengthened forms of the classical inequalities between the volume of a convex body and its projection measures (Quermassintegrale). The relation between the Busemann-Petty centroid inequality and the Petty projection inequality is shown to be similar to the relation that exists between the Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry. Some mixed integral inequalities are derived similar in spirit to inequalities obtained by Chakerian and others.