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Some isoperimetric inequalities with application to the Stekloff problem. (English) Zbl 1155.26019

Let \(\Omega\in\mathbb{R}^n\) be a measurable set. Consider its moments of inertia with respect to the planes \(x_k=0\), i.e. \(J_k(\Omega)=\int_{\Omega}x_k^2\,d\mathbf{x}\), \(k=1,\dots,n\). Consider also te moments of inertia of the boundary \(\partial\Omega\) with respect to these planes, i.e., \(I_k(\Omega)=\int_{\partial\Omega}x_k^2\,ds\), \(k=1,\dots,n\). In the paper the products \(J(\Omega)=\prod_{k=1}^nJ_k(\Omega)\) and \(I(\Omega)=\prod_{k=1}^nI_k(\Omega)\) are studied. They are minimized in the class \(\mathcal{O}\) of bounded measurable subsets of \(\mathbb{R}^n\) with a prescribed volume.
Let \(B\in\mathcal{O}\) be a ball centered at the origin and \(E\in\mathcal{O}\) an ellipsoid symmetric with respect to the planes \(x_k=0\), \(k=1,\dots,n\). Two isoperimetric inequalities are proved: \(J(\Omega)\geq J(E)\) for all \(\Omega\in\mathcal{O}\) and \(I(\Omega)\geq I(B)\) for all convex sets \(\Omega\in\mathcal{O}\) (it is shown that in the planar case convexity assumption can be dropped). The motivation for studying the products of moments of inertia is presented in the Introduction. The main results are applied to the minimization of the inertia matrix and to the Stekloff eigenvalue problem.

MSC:

26D15 Inequalities for sums, series and integrals
52A40 Inequalities and extremum problems involving convexity in convex geometry
35P15 Estimates of eigenvalues in context of PDEs
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