Local Optimality of the d-cube for Laplacian Eigenvalues
Abstract
For each d ≥ 2 we prove that the unit d-cube □d ⊂ ℝd is, on the class 𝒫 of convex polytopes combinatorially equivalent to a d-cube, (i) a strict local minimizer of the scale-invariant isoperimetric ratio I(Ω) = P(Ω)d/(|Ω|d−1), with P the perimeter, (ii) a strict local minimizer of the volume-normalized first Dirichlet eigenvalue λ̅(Ω) = |Ω|2/dλ1(Ω), and (iii) a strict local maximizer of the volume-normalized first Neumann eigenvalue μ̅(Ω) = |Ω|2/dμ1(Ω). 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 π, rational coefficients, and a few universal series-constants. The first Neumann eigenvalue is not differentiable at □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, λ̅, and μ̅ does not hold in high dimensions for a particular multilinear perturbation that leaves 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 ≥ 3.
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BibTeX
@misc{osting2026local,
title = {{Local Optimality of the d-cube for Laplacian Eigenvalues}},
author = {Braxton Osting and Robert Viator},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30463},
eprint = {2610.30463},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30463},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Local Optimality of the d-cube for Laplacian Eigenvalues
AU - Braxton Osting
AU - Robert Viator
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30463
UR - https://arxiv.org/abs/2610.30463
AB - For each d ≥ 2 we prove that the unit d-cube □_d ⊂ ℝ^d is, on the class 𝒫 of convex polytopes combinatorially equivalent to a d-cube, (i) a strict local minimizer of the scale-invariant isoperimetric ratio I(Ω) = P(Ω)^d/(|Ω|^d−1), with P the perimeter, (ii) a strict local minimizer of the volume-normalized first Dirichlet eigenvalue λ̅(Ω) = |Ω|^2/dλ_1(Ω), and (iii) a strict local maximizer of the volume-normalized first Neumann eigenvalue μ̅(Ω) = |Ω|^2/dμ_1(Ω). 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 π, rational coefficients, and a few universal series-constants. The first Neumann eigenvalue is not differentiable at □_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, λ̅, and μ̅ does not hold in high dimensions for a particular multilinear perturbation that leaves 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 ≥ 3.
ER -