Blaschke-Santaló and Mahler inequalities for the first eigenvalue of the Dirichlet Laplacian (Q2832000)

The \(\lambda_1\)-product functional associates with a given convex body \(K\subset {\mathbb R}^n\) the quantity \(\lambda_1(K)\lambda_1(K^o)\), where \(K^o\) is the polar body of \(K\) and \(\lambda_1(\cdot)\) is the first eigenvalue of the Dirichlet Laplacian.NEWLINENEWLINEThe first result in this paper, Theorem~1, shows that the minimum of the \(\lambda_1\)-product functional in the class of centrally symmetric convex bodies is attained by balls: when \(K\) is centrally symmetric and \(B\) is a ball, there holds NEWLINE\[NEWLINE \lambda_1(K)\lambda_1(K^o)\geq \lambda_1(B)\lambda_1(B^o). NEWLINE\]NEWLINE The proof of this inequality follows from the classical Blaschke-Santaló's and Faber-Krahn inequalities.NEWLINENEWLINEThe second main result is Theorem~9, where it is shown that the supremum of the planar functional NEWLINE\[NEWLINE \inf_{T\in D_2} \lambda_1(T(K))\lambda_1(T(K)^o) NEWLINE\]NEWLINE in \({\mathcal K}_{\#}^2\) is achieved by the square. Here \({\mathcal K}_{\#}^2\) is the set of unconditional bodies (i.e. those symmetric with respect to all coordinate hyperplanes of a fixed frame) and \(D_2\) is the class of invertible diagonal transformations of \({\mathbb R}^2\). As the authors show in Remark~3, the restriction on the space is necessary since the supremum of the \(\lambda_1\)-product functional is \(+\infty\) in the class of convex bodies and in the one of centrally symmetric convex bodies.NEWLINENEWLINEFor the proof of Theorem~9, the authors first show that a sufficient condition for the square to be a minimizer is that it solves the problem NEWLINE\[NEWLINE \sup\{[\lambda_1(\Omega)|\Omega|]:\Omega\in {\mathcal O}\}, NEWLINE\]NEWLINE where \({\mathcal O}\) is the class of convex axisymmetric octagons having their vertices lying on the axes at the same distance. The proof of the latter result, Theorem~12, is given by a hybrid method involving both theoretical and numerical tools.