| Abstract: |
| Given any open, bounded set $\Omega \subset \mathbb{R}^N$, Cheeger inequality states that
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\[
\mathcal{J}_{1,p}[\Omega] := \frac{\lambda_p(\Omega)}{h(\Omega)^p} \ge \frac {1}{p^p},
\]
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where $h(\Omega)$ denotes the Cheeger constant of $\Omega$ and $\lambda_{1,p}(\Omega)$ the first Dirichlet eigenvalue of the $p$-Laplacian.
A natural question is whether the shape functional $\mathcal{J}_{1,p}[\,\cdot\,]$ admits minimizers in some suitable class of sets. Denoting with $\mathcal{K}_N$ the class of \emph{convex} subsets of $\mathbb{R}^N$, Parini proved that the shape functional $\mathcal{J}_{1,2}[\,\cdot\,]$ admits minimizers in $\mathcal{K}_2$. Recently Briani, Buttazzo and Prinari proved existence in $\mathcal{K}_2$ for the more general shape functional $\mathcal{J}_{1,p}[\,\cdot\,]$, and conjectured existence of minimizers in $\mathcal{K}_N$ to hold true regardless of the dimension $N$.
Together with Aldo Pratelli, we prove this conjecture. Our proof exploits a criterion proved by Ftouhi paired with some cylindrical estimate on the Cheeger constant of $(N+1)$-dimensional cylinders $\Omega \times [0,L]$ in terms of the Cheeger constant of the cross-section $\Omega$. |
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