Sections of convex bodies

The generalized Busemann-Petty problem asks: If the volume of i-dimensional central section of a centrally symmetric convex body in R(n) is smaller than that of another such body, is the volume of the body also smaller? It is proved that the answer is negative if 2 < i < n. The case of a 2-dimensional section remains open. The proof uses techniques in functional analysis and Radon transforms on Grassmannians. It also requires the notion of an i-intersection body which generalizes the notion of an intersection body. Inequalities among the volumes of projection bodies, polar projection bodies and their central sections are proved. They are related to the maximal slice problem.