*Published Paper*

**Inserted:** 24 mar 2020

**Last Updated:** 6 nov 2023

**Journal:** Ann. Fenn. Math.

**Volume:** 46

**Number:** 2

**Pages:** 1071--1087

**Year:** 2021

**Doi:** 10.5186/aasfm.2021.4666

In the published paper in the definition (2.2) of the index $\zeta(\Omega)$ the rescaling factor is missing (as otherwise the quantitative inequality (2.5) would be false as written). Consequently, in the proof of Theorem 2.1 and in the following example several renormalization constants are missing. The preprint contains the correct definition and proofs.

**Abstract:**

We provide a quantitative lower bound to the Cheeger constant of a set $\Omega$ in both the Euclidean and the Gaussian settings in terms of suitable asymmetry indexes. We provide examples which show that these quantitative estimates are sharp.

**Keywords:**
Cheeger sets, Cheeger constant, quantitative inequalities

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