How far apart can the projection of the centroid of a convex body and the centroid of its projection be?

TL;DR

This paper establishes a universal bound on the distance between the projection of a convex body's centroid and the centroid of its projection, showing it is at most approximately 0.2016 times the body's width, with the bound being asymptotically sharp.

Contribution

The paper proves a sharp universal constant bound on the distance between two specific centroids related to convex bodies and their projections.

Findings

The constant D is approximately 0.2016.

The bound is asymptotically sharp.

The result holds for all convex bodies in any dimension.

Abstract

We show that there is a constant $D \approx 0.2016$ such that for every $n$, every convex body $K\subset \mathbb R^n$, and every hyperplane $H\subset \mathbb R^n$, the distance between the projection of the centroid of $K$ onto $H$ and the centroid of the projection of $K$ onto $H$ is at most $D$ times the width of $K$ in the direction of the segment connecting the two points. The constant $D$ is asymptotically sharp.