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arXiv · 2112.13590

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

René Brandenberg·Katherina von Dichter·Bernardo González Merino
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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.

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René Brandenberg, Katherina von Dichter, Bernardo González Merino. 2021-12-27. Tightening and reversing the arithmetic-harmonic mean inequality for symmetrizations of convex sets. https://arxiv.org/abs/2112.13590

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