The paper studies convex bodies and star bodies in $\Bbb R^n$ by using Radon transforms on Grassmann manifolds, $p$-cosine transforms on the unit sphere, and convolutions on the rotation group of $\Bbb R^n$. It presents dual mixed volume characterizations of $i$-intersection bodies and $L_p$-balls which are related to certain volume inequalities for cross sections of convex bodies. It considers approximations of special convex bodies by analytic bodies and various finite sums of ellipsoids which preserve special geometric properties. Convolution techniques are used to derive formulas for mixed volumes, mixed surface measures, and $p$-cosine transforms. They are also used to prove characterizations of geometric functionals, such as surface area and dual quermassintegrals.