Isomorphic properties of Intersection bodies

TL;DR

This paper investigates the geometric properties of k-intersection bodies and generalized k-intersection bodies, establishing bounds on their Banach-Mazur distance to Euclidean balls and providing volumetric estimates.

Contribution

It introduces new isomorphic properties of these bodies and generalizes existing results on their distances to Euclidean balls.

Findings

Banach-Mazur distance of convex k-intersection bodies to Euclidean balls is bounded by a k-dependent constant.

Provides volumetric estimates for k-intersection bodies.

Extends known results of Hensley and Borell to broader classes of intersection bodies.

Abstract

We study isomorphic properties of two generalizations of intersection bodies, the class of k-intersection bodies and the class of generalized k-intersection bodies. We also show that the Banach-Mazur distance of the k-intersection body of a convex body, when it exists and it is convex, with the Euclidean ball, is bounded by a constant depending only on k, generalizing a well-known result of Hensley and Borell. We conclude by giving some volumetric estimates for k-intersection bodies.