Area-Refined Spectral Gap Bounds for Convex Domains

Definitions and mathematical statement

The supplied record does not specify the operator or boundary conditions defining the eigenvalues. The following is therefore a conditional formulation under the Dirichlet Laplacian interpretation; that interpretation requires confirmation from the report.

Let $K\subset{\mathbb{R}}^2$ be a nonempty bounded open convex set, and write \[ A=|K|,\qquad d=\sup\{|x-y|:x,y\in K\}. \] Boundedness, openness, and nonempty interior are explicit assumptions of this conditional formulation. Let $\lambda_1(K)\leq\lambda_2(K)\leq\cdots$ denote the eigenvalues, counted with multiplicity, of $-\Delta$ on $K$ with homogeneous Dirichlet boundary conditions, and define \[ \gamma(K)=\lambda_2(K)-\lambda_1(K). \] Let $B_1$ be a disk of area $1$, and define $\gamma(B_1)$ using the same operator and boundary conditions.

The principal question recorded in the corrected source statement is whether every such $K$ satisfies \[ \gamma(K)\geq \frac{12\pi^2\gamma(B_1)}{ 3\pi^3d^2+ \left[4\gamma(B_1)-3\pi^3\right] \sqrt{d^4-\frac{16A^2}{\pi^2}} }. \] It also asks whether equality among bounded domains occurs precisely for disks. The record additionally asserts an asymptotic equality possibility for domains converging to an infinite strip. This latter clause is not a precise quantified assertion as supplied: the sequence of domains, its normalization, the notion of convergence, and the measure of asymptotic equality are unspecified. An infinite strip itself is outside the bounded-domain formulation above.

The original statement also records the separate fundamental-gap conjecture \[ \gamma(K)\geq\frac{3\pi^2}{d^2}. \] The supplied dated literature assessment attributes a proof of this first conjecture for convex domains to Andrews and Clutterbuck, while describing the stronger area-corrected conjecture as not established in general. These are supplied curation claims, not independently verified conclusions here.

The original extraction used $\sqrt{d^4-16A^2}$ in the area-corrected denominator. The supplied verification explicitly identifies this as an extraction error and identifies $\sqrt{d^4-16A^2/\pi^2}$ as the expression in Conjecture 2 of the report. The displayed principal question follows that supplied correction.

Short English statement

Under an explicitly conditional Dirichlet Laplacian interpretation, the question asks for an area-dependent lower bound on the spectral gap of every bounded planar convex domain, with equality precisely for disks. The claimed limiting equality for infinite strips needs a specified limiting procedure. The supplied dated assessment labels the record partially solved, distinguishing the fundamental-gap claim from the stronger area-corrected question.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001155$, title: Area-Refined Spectral Gap Bounds for Convex Domains; exact source problem code: catalogue entry. The record attributes the questions to a 2009 Oberwolfach workshop report; no individual proposer is supplied.
• Low Eigenvalues of Laplace and Schrödinger Operators (2009), Oberwolfach Reports, pp. 355--428, DOI: 10.4171/owr/2009/06. Supplied report and literature-source link: https://doi.org/10.4171/owr/2009/06. The supplied verification identifies the corrected area-refined statement as Conjecture 2.
• Andrews--Clutterbuck, fundamental gap: supplied literature-source link https://arxiv.org/abs/1006.1686. The supplied assessment attributes the convex-domain fundamental-gap result to these authors.
• Literature checked at: 2026-08-21. This exact date and the assessment that the fundamental-gap conjecture is proved for convex domains while the stronger area-corrected conjecture is not established in general are supplied dated curation, not independently verified here. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and its status partially_solved. These are supplied metadata labels; this draft does not independently certify them.