Relating Symmetrizations of Convex Bodies: Once More the Golden Ratio

TL;DR

This paper characterizes when the harmonic mean of a Minkowski centered convex set and its reflection is contained in its arithmetic mean, revealing a connection to the golden ratio and a special pentagon shape.

Contribution

It establishes a precise condition involving the Minkowski asymmetry and introduces the 'golden house' pentagon as the extremal shape.

Findings

Harmonic mean containment occurs iff asymmetry ≤ golden ratio

The extremal shape is a pentagon called the golden house

Connects symmetrization, convex geometry, and the golden ratio

Abstract

We show that for any Minkowski centered planar convex compact set $C$ the Harmonic mean of $C$ and $-C$ can be optimally contained in the arithmetic mean of the same sets if and only if the Minkowski asymmetry of $C$ is at most the golden ratio $(1+\sqrt{5})/2 \approx 1.618$. Moreover, the most asymmetric such set that is (up to a linear transformation) a special pentagon, which we call the golden house.