Let \(\Omega\subset \mathbb{R}^n\) be any open set with finite (Lebesgue) measure. Its Cheeger constant is defined as \[ h(\Omega)=\inf \left\{\tfrac{P(E)}{|E|}, E\subset \Omega\right\} \] (where \(P(E)\) is the perimeter and \(|E|\) the measure). Let us denote by \(B_\Omega\) the ball, centered at the origin, with the same measure as \(\Omega\). The isoperimetric inequality \(h(\Omega) \geq h(B_\Omega)\) is well known. A quantitative improvement has been given by Figalli, Maggi, Pratelli in [A. Figalli et al., Proc. Am. Math. Soc. 137, No. 6, 2057–2062 (2009; Zbl 1168.39008)], namely \[ \frac{h(\Omega) - h(B_\Omega)}{h(B_\Omega)} \geq c(n) \alpha^2(\Omega) \] where \(\alpha(\Omega)\) is the so-called Fraenkel asymmetry and \(c(n)\) a dimensional constant.
In this paper, the authors improve a little bit this quantitative inequality by replacing the Fraenkel asymmetry by the stronger measure of asymmetry, known as the Riesz asymmetry index defined by: \[ \zeta(\Omega)=\int_{B_\Omega} \frac{dx}{|x|}-\max_{y\in \mathbb{R}^n} \int_\Omega \frac{dx}{|x-y|}. \] They prove the following inequality \[ \frac{h(\Omega) - h(B_\Omega)}{h(B_\Omega)} \geq c(n) \zeta(\Omega). \] In the second part of the paper, the authors give the same kind of result in the Gaussian setting (that seems to be new), replacing the Lebesgue measure by the Gaussian measure and the perimeter by the Gaussian perimeter. In that context, the quantitative inequality is given in terms of the Gaussian Fraenkel asymmetry. They also provide another inequality in terms of a different index involving the barycenter. They give, in each case, examples proving that the inequalities are sharp.