Local Optimality of the d-cube for Laplacian Eigenvalues

Abstract

For each $d\ge 2$ we prove that the unit $d$-cube ${\square }_d\subset {\mathbb{R}}^d$ is, on the class $\mathcal{P}$ of convex polytopes combinatorially equivalent to a $d$-cube, (i) a strict local minimizer of the scale-invariant isoperimetric ratio $I(\Omega )=\frac{P(\Omega {)}^d}{|\Omega {|}^{d-1}}$, with $P$ the perimeter, (ii) a strict local minimizer of the volume-normalized first Dirichlet eigenvalue $\bar{\lambda }(\Omega )=|\Omega {|}^{2/d}{\lambda }_1(\Omega )$, and (iii) a strict local maximizer of the volume-normalized first Neumann eigenvalue $\bar{\mu }(\Omega )=|\Omega {|}^{2/d}{\mu }_1(\Omega )$. The method is a shape-derivative Hessian analysis adapted to the hyperoctahedral symmetry of the cube. In symmetry-adapted coordinates on the parameter space --- the decomposition into irreducible representations of the hyperoctahedral group --- each Hessian is block-diagonal, with four scalar eigenvalues, which we give in closed form in $\pi$, rational coefficients, and a few universal series-constants. The first Neumann eigenvalue is not differentiable at ${\square }_d$, since its lowest cluster splits under perturbation. The Neumann case therefore works with a smooth auxiliary functional, whose Hessian is computed by second-order perturbation theory of the cluster; the coupling to modes outside the cluster enters at the same order. Finally, we show that extremality of $I$, $\bar{\lambda }$, and $\bar{\mu }$ does not hold in high dimensions for a particular multilinear perturbation that leaves $\mathcal{P}$ by breaking convexity. All results are local; we discuss why a global statement, available for quadrilaterals (Endo and Osting, 2026), appears out of reach for $d\ge 3$.

arXiv comment32 pages, 2 figures