Tightening and reversing the arithmetic-harmonic mean inequality for symmetrizations of convex sets

TL;DR

This paper investigates symmetrizations of convex sets, establishing conditions to reverse their inclusion relations and tightening existing inequalities, with stability analysis near the simplex.

Contribution

It determines dilatation factors to reverse symmetrization containments and improves bounds on inequalities among convex set symmetrizations.

Findings

Dilatation factors depend on set asymmetry

Reversal of symmetrization containments achieved

Stability results near the simplex

Abstract

This paper deals with four symmetrizations of a convex set $C$: the intersection, the harmonic and the arithmetic mean, and the convex hull of $C$ and $-C$. A well-known result of Firey shows that those means build up a subset-chain in the given order. On the one hand, we determine the dilatation factors, depending on the asymmetry of $C$, to reverse the containments between any of those symmetrizations. On the other hand, we tighten the relations proven by Firey and show a stability result concerning those factors near the simplex.