One of the fundamental results regarding intersection bodies is Busemann’s theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is natural to ask how much of convexity is guaranteed under the intersection body operation. In this paper we provide several results about the (strict) improvement of convexity for intersection bodies of symmetric bodies of r...

We construct symmetric convex bodies that are not intersection bodies, but all of their central hyperplane sections are intersection bodies. This result extends the studies by Weil in the case of zonoids and by Neyman in the case of subspaces of Lp.

For convex bodies K; K 0 and a translation in n-dimensional Euclidean space, let K _ K 0 be the convex hull of the union of K and K 0. Let F be a geometric functional on the space of all convex bodies. We consider special families (r) r>0 of measures on the translation group T n such that the limit lim r!1 Z Tn F (K _ K 0) dd r () exists and can be expressed in terms of K and K 0. The functiona...

Let Ik be the class of convex k-intersection bodies in R n (in the sense of Koldobsky) and Im k be the class of convex origin-symmetric bodies all of whose m-dimensional central sections are k-intersection bodies. We show that 1) Im k 6⊂ I m+1 k , k + 3 ≤ m < n, and 2) Il 6⊂ Ik, 1 ≤ k < l < n − 3.