On a class of Cheeger inequalities

We study a general version of the Cheeger inequality by considering the shape functional $\mathcal{F}_{p,q}(Ω)=λ_p^{1/p}(Ω)/λ_q(Ω)^{1/q}$. The infimum and the supremum of $\mathcal{F}_{p,q}$ are studied in the class of all domains $Ω$ of $\mathbb{R}^d$ and in the subclass of convex domains. In the latter case the issue concerning the existence of an optimal domain for $\mathcal{F}_{p,q}$ is discussed.

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