Volume inequalities for sets associated with convex bodies

This paper deals with inequalities for the volume of a convex body and the volume of the projection body, the Lp-centroid body, and theyr polars. Examples are the Blaschke-Santalò inequality, the Petty and Zhang projection inequalities, the Busemann-Petty inequality. Other inequalities of the same type are still at the stage of conjectures. The use of special continuous movements of convex bodies provides a general approach to this subject. A family of inequalities, depending on a parameter p and proved by Lutwak for p=1 and p=2 , is obtained