Position of the centroid of a planar convex body

TL;DR

The paper proves that for any planar convex body, the centroid lies within a specific scaled and centered affine-regular hexagon inscribed in it, establishing a precise ratio that cannot be improved.

Contribution

It introduces a new geometric bound relating the centroid of a convex body to an inscribed affine-regular hexagon, with a proven optimal ratio.

Findings

Centroid is within a scaled affine-regular hexagon inscribed in the convex body.

The ratio of 4/21 for the homothety is optimal.

The centroid's position relative to the inscribed hexagon is precisely characterized.

Abstract

It is well known that any planar convex body $A$ permits to inscribe an affine-regular hexagon $H_A$. We prove that the centroid of $A$ belongs to the homothetic image of $H_A$ with ratio $\frac{4}{21}$ and the center in the center of $H_A$. This ratio cannot be enlarged.