8 research outputs found
Isomorphic properties of Intersection bodies
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
The complex Busemann-Petty problem on sections of convex bodies
The complex Busemann-Petty problem asks whether origin symmetric convex
bodies in \C^n with smaller central hyperplane sections necessarily have
smaller volume. We prove that the answer is affirmative if and
negative if Comment: 18 page
Complex Intersection Bodies
We introduce complex intersection bodies and show that their properties and
applications are similar to those of their real counterparts. In particular, we
generalize Busemann's theorem to the complex case by proving that complex
intersection bodies of symmetric complex convex bodies are also convex. Other
results include stability in the complex Busemann-Petty problem for arbitrary
measures and the corresponding hyperplane inequality for measures of complex
intersection bodies
Positive definite distributions and normed spaces
AbstractWe answer a question of Alex Koldobsky. We show that for each −∞<p<2 and each n⩾3−p there is a normed space X of dimension n which embeds in Ls if and only if −n<s⩽p
Sections of convex bodies with symmetries
In this paper we study how certain symmetries of convex bodies affect their geometric properties. In particular, we consider the impact of symmetries generated by the block diagonal subgroup of orthogonal transformations, generalizing complex and quaternionic convex bodies. We conduct a systematic study of sections of bodies with symmetries of this type, with the emphasis on problems of the Busemann-Petty type and hyperplane inequalities. The main role belongs to the class of intersection bodies with symmetries. © 2014 Elsevier Inc