{
  "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 $I(Ω) = \\tfrac{P(Ω)^d}{|Ω|^{d-1}}$, with $P$ the perimeter, (ii) a strict local minimizer of the volume-normalized first Dirichlet eigenvalue $\\barλ(Ω) = |Ω|^{2/d}λ_1(Ω)$, and (iii) a strict local maximizer of the volume-normalized first Neumann eigenvalue $\\barμ(Ω) = |Ω|^{2/d}μ_1(Ω)$. 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 $\\square_d$, 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 $I$, $\\barλ$, and $\\barμ$ 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 $d \\ge 3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30463",
  "author_total": 2,
  "authors": [
    {
      "name": "Braxton Osting"
    },
    {
      "name": "Robert Viator"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aw7qxqf2f1xaj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30463",
  "fulltext_url": null,
  "id": "aw7qxqf2f1xaj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30463",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7223410670",
  "pdf_url": "https://arxiv.org/pdf/2610.30463",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aw7qxqf2f1xaj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30463",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30463"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Local Optimality of the d-cube for Laplacian Eigenvalues",
  "title_text": "Local Optimality of the d-cube for Laplacian Eigenvalues",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/aw7qxqf2f1xaj",
  "venue": "arXiv",
  "versions": []
}