Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain (Q1272838)

Let \(M\) be a compact connected domain in \(\mathbb{R}^n\) and let \(\Delta\) be the usual Laplace operator. Impose Dirichlet boundary conditions and let \(0<\lambda_1<\lambda_2\leq\lambda_3\dots\) be the associated eigenvalues; the ground state \(\lambda_1\) has simple multiplicity. The authors introduce a weighted Cheeger constant and estimate the gap \(\lambda_2-\lambda_1\) in terms of this constant. If the domain satisfies an interior rolling sphere condition, the authors give an estimate on the weighted Cheeger constant in terms of the rolling sphere radius, the volume, a bound on the principal curvatures of the boundary, and the dimension.