**Beiträge zur Algebra und Geometrie / Contributions to Algebra and GeometryVol. 40, No. 1, pp. 163-183 (1999)
**

#
Hyperplane Sections of Convex Bodies in Isotropic Position

##
Matthieu Fradelizi

Universite de Marne-la-Vallee, Equipe d'Analyse et de Mathematiques Appliquees, Cite Descartes, 5 Bd Descartes, Champs sur Marne, 77454 Marne-la-Vallee Cedex 2, France, e-mail: fradeliz@math.univ-mlv.fr

**Abstract:** We prove sharp inequalities showing 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, does not depend much of the hyperplane. This generalizes a result of K. Ball to non-symmetric convex bodies. For the sections through its centroid, the extremal bodies are the same as in the symmetric case (cylinders and double-cones). For the sections of maximal volume parallel to a fixed hyperplane, the extremal bodies are cylinders and cones. We deduce that the cross-section body and the intersection body of a convex body with respect to its centroid are close to its Binet ellipsoid; we generalize results of J. Bourgain on maximal functions to the case of non-symmetric convex bodies and we prove that some equivalences between different forms of the hyperplane conjecture still hold in the non-symmetric case.

**Keywords:** volume, section, centroid, isotropic, ellipsoid, inertia, hyperplane, intersection body, cross-section body, maximal function

**Classification (MSC91):** 52A20, 52A40

**Full text of the article:**

[Previous Article] [Next Article] [Contents of this Number]