Convolutions, Transforms, and Convex Bodies

The paper studies convex bodies and star bodies in Rⁿ by usingRadon transforms on Grassmann manifolds, p-cosine transformson the unit sphere, and convolutions on the rotation group ofRⁿ. It presents dual mixed volume characterizations of i-intersectionbodies and L_p-balls which are related to certain volume inequalitiesfor cross sections of convex bodies. It considers approximationsof special convex bodies by analytic bodies and various finitesums of ellipsoids which preserve special geometric properties.Convolution techniques are used to derive formulas for mixedvolumes, mixed surface measures, and p-cosine transforms. Theyare also used to prove characterizations of geometric functionals,such as surface area and dual quermassintegrals. 1991 MathematicsSubject Classification: 52A20, 52A40.