Abstract
This paper deals with inequalities for the volume of a convex body and the volume of the projection body, the L-centroid body, and their polars. Examples are the Blaschke-Santalo 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 ≥ 1 and proved by Lutwak for p = 1 and p = 2, is obtained.
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