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Description
Given a set $\Omega\subset \mathbb{R}^N$, the Cheeger constant is a purely geometrical quantity defined as the infimum
$$
h(\Omega):= \inf \left\{\,\frac{P(E)}{|E|}\,:\, E\subset \Omega,\, |E|>0 \,\right\}.
$$
Despite seeming unassuming, it pops up in many contexts that apparently have nothing in common. To name a few, under some mild regularity assumptions on $\Omega$: bounds on the first Dirichlet eigenvalue of the $p$-Laplacian; existence of sets in $\Omega$ or of graphs over $\Omega$ with prescribed curvature; threshold of vertical load that a flat membrane can sustain before breaking; image reconstruction and denoising. The constant of the unit square has even been a tool in a elementary proof of the Prime Number Theorem!
Given the numerous applications, it is important being able to explicitly compute the constant. This is in general a hard task: a telltale sign is that we do not know the exact value of the constant of the unit cube in dimension $N\ge 3$. The computation is (theoretically) feasible for a large class of Jordan domains in the plane [LNS] or in very special cases in general dimension.
If unable to compute the constant, it would be at least desirable to obtain bounds on it: in [LNS] we proved bounds via interior approximations of the set for $2$d domains on which, at least on a theoretical level, the constant can be found by solving an algebraic equation; in [JS] a quantitative inequality for the Cheeger constant has been proved in terms of the Riesz asymmetry; in [BPS] bounds of the constant for cylindrical domains $\Omega=\omega\times[0,L]$ have been shown in terms of the constant of the cross-section $\omega$.
[BPS] G. Buttazzo, A. Pratelli, and G. Saracco. Upper and lower bounds on the first Dirichlet eigenvalue of the $p$-Laplacian in cylindrical domains, and existence of minimizers of a shape optimization problem. Forthcoming.
[JS] V. Julin and G. Saracco. “Quantitative lower bounds to the Euclidean and the Gaussian Cheeger constants.” In: Ann. Fenn. Math. 46.2 (2021), pp. 1071–1087.
[LNS] G. P. Leonardi, R. Neumayer, and G. Saracco. “The Cheeger constant of a Jordan domain without necks.” In: Calc. Var. Partial Differential Equations 56.6 (2017), p. 164.
