104 research outputs found
Estimates for measures of sections of convex bodies
A estimate in the hyperplane problem with arbitrary measures has
recently been proved in \cite{K3}. In this note we present analogs of this
result for sections of lower dimensions and in the complex case. We deduce
these inequalities from stability in comparison problems for different
generalizations of intersection bodies
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
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
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
Measure comparison and distance inequalities for convex bodies
We prove new versions of the isomorphic Busemann-Petty problem for two different measures and show how these results can be used to recover slicing and distance inequalities. We also prove a sharp upper estimate for the outer volume ratio distance from an arbitrary convex body to the unit balls of subspaces of
Inequalities of the Kahane–Khinchin type and sections of L p
International audienceWe extend Kahane-Khinchin type inequalities to the case p > -2. As an application we verify the slicing problem for the unit balls of finite-dimensional spaces that embed in L-p, p, > -2
- …