Hyperplane sections of convex bodies in isotropic position
type: Item
instance of: scholarly article
title: Hyperplane sections of convex bodies in isotropic position
zbMATH Open document ID: 0966.52004
author: Matthieu Fradelizi
published in: Beiträge zur Algebra und Geometrie
publication date: 2001-01-09T00:00:00Z
full work available at URL: https://eudml.org/doc/232153
full work available at URL: 13.html
review text: The author obtains sharp inequalities bounding above and below \[ \int_K\phi(\langle x,u\rangle) dx \] where \(\phi\) is a convex function (sometimes assumed to be even), \(u\) is a unit vector and \(K\) ranges over the set of convex bodies whose centroids are at the origin. Extremum values are also obtained. As a consequence of these inequalities the author presents important geometric results: He proves that the volume of the hyperplane sections of a convex body in isotropic position through its centroid, as well as those of maximal volume in a fixed direction, do not depend much on the hyperplane (they are almost constant). This is a generalization of a result of K. Ball to non-symmetric convex bodies. The extremal bodies are cylinders and double-cones (for the sections through their centroids) and cylinders and cones (for the sections of maximal volume parallel to a fixed hyperplane). He also proves that the cross-section body and the intersection body of a convex body with respect to its centroid are close to its Binet ellipsoid (the polar of its inertia ellipsoid). This generalizes results of J. Bourgain to the case of non-symmetric convex bodies.
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 1549695
zbMATH Keywords: volume
zbMATH Keywords: section
zbMATH Keywords: centroid
zbMATH Keywords: isotropic
zbMATH Keywords: ellipsoid
zbMATH Keywords: inertia
zbMATH Keywords: hyperplane
zbMATH Keywords: intersection body
zbMATH Keywords: cross-section body
zbMATH Keywords: maximal function
zbMATH Keywords: inequalities
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1549695