Hyperplane sections of convex bodies in isotropic position (Q1591725)

The author obtains sharp inequalities bounding above and below \[ \int_K\phi(\langle x,u\rangle) dx \] where \(\phi\) is a convex function (sometimes assumed to be even), \(u\) is a unit vector and \(K\) ranges over the set of convex bodies whose centroids are at the origin. Extremum values are also obtained. As a consequence of these inequalities the author presents important geometric results: He proves that the volume of the hyperplane sections of a convex body in isotropic position through its centroid, as well as those of maximal volume in a fixed direction, do not depend much on the hyperplane (they are almost constant). This is a generalization of a result of K. Ball to non-symmetric convex bodies. The extremal bodies are cylinders and double-cones (for the sections through their centroids) and cylinders and cones (for the sections of maximal volume parallel to a fixed hyperplane). He also proves that the cross-section body and the intersection body of a convex body with respect to its centroid are close to its Binet ellipsoid (the polar of its inertia ellipsoid). This generalizes results of J. Bourgain to the case of non-symmetric convex bodies.