| Convex bodies, star bodies and volume inequalities in {dollar}{lcub}bf R{rcub}sp{lcub}n{rcub}{dollar} are investigated by using Radon transforms on the Grassmannians, convolutions on the rotation group and elliptic differential operators on the unit sphere. Analytic and geometric characterizations of i-intersection bodies and {dollar}Lsb{lcub}p{rcub}{dollar} balls are given. They yield inequalities among the volumes of bodies and their sections and projections. Affine isoperimetric inequalities associated with these bodies are proved.; A simple-sounding problem states that if the i-dimensional central section of a symmetric convex body K in {dollar}{lcub}bf R{rcub}sp{lcub}n{rcub}{dollar} has smaller volume than that of another symmetric convex body, does K itself have smaller volume? It is proved that this problem has a positive answer if K is an i-intersection body, but in general, it has a negative answer when i {dollar}>{dollar} 2.; Ellipsoidal decompositions of symmetric convex bodies and star bodies are considered. It is shown that every symmetric convex body is the Hausdorff limit of Blaschke sums of ellipsoids. It is proved that i-intersection bodies and {dollar}Lsb{lcub}p{rcub}{dollar} balls all can be decomposed into ellipsoids in the sense of the sum of radial functions (or norms) with suitable powers.; Elliptic differential operators associated with convex bodies are discussed. They are applied to show the openness of curvature functions, and to prove a uniqueness-existence theorem about deformations of hypersurfaces in differential geometry. A short approach to the Christoffel problem is given by using techniques of convolutions and Green's function on the unit sphere.; New integral-geometric formulas are presented. They involve dual quermass-integrals of star bodies, and are dual to Blaschke-Chern's formula and Crofton-Santalo's formula in integral geometry. |
