Isoperimetric inequalities and identities for \(k\)-dimensional cross-sections of convex bodies (Q2276762)

We obtain an estimate for the volume of a convex body M in \({\mathbb{R}}^ n\) by means of the areas of its slices by k-dimensional planes through the origin. When M is an ellipsoid centered at the origin this estimate reduces to an integral formula of Fürstenberg and Tzkoni. We show that the Fürstenberg-Tzkoni formula is valid only for such M and hence characterizes ellipsoids. The case \(k=n-1\) is due to H. Busemann and plays a crucial role in the analysis. We give a corresponding estimate for convex bodies in \({\mathbb{C}}^ n\) and an integral formula which characterizes complex ellipsoids. We also prove the affine invariance of related affine quermassintegrals.