On the volume of sections of a convex body by cones

TL;DR

This paper proves that sections of convex bodies through the centroid are nearly symmetric in volume in small codimensions, providing insights into convex intersection bodies and addressing a problem posed by Meyer and Reisner.

Contribution

It establishes volume symmetry properties of convex body sections through the centroid in small codimensions, advancing understanding of convex intersection bodies.

Findings

Sections through the centroid are nearly symmetric in volume in small codimensions

Provides a positive answer to a problem by Meyer and Reisner on convex intersection bodies

Enhances understanding of volume distribution in convex geometry

Abstract

Let $K$ be a convex body in $\mathbb R^n$. We prove that in small codimensions, the sections of a convex body through the centroid are quite symmetric with respect to volume. As a consequence of our estimates we give a positive answer to a problem posed by M. Meyer and S. Reisner regarding convex intersection bodies.