Relating Symmetrizations of Convex Bodies: Once More 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.