Validity of the fundamental gap estimate with Gaussian-metric diameter

Determine whether the lower bound for the fundamental gap of the Ornstein–Uhlenbeck operator L_μ = Δ − ⟨x, ∇⟩ with Dirichlet boundary conditions on a strictly convex domain Ω ⊂ ℝ^n, namely λ_{2,μ}(Ω) − λ_{1,μ}(Ω) ≥ \bar{λ}_2(D) − \bar{λ}_1(D) where D is the Euclidean diameter of Ω and \bar{λ}_1(D), \bar{λ}_2(D) are the first two Dirichlet eigenvalues of the one-dimensional operator −d^2/ds^2 + (1/4)s^2 on (−D/2, D/2), remains valid when D is replaced by D_G, the diameter of Ω measured with respect to the Gaussian metric.

Background

The main theorems of the paper establish a sharp lower bound for the fundamental gap of the Ornstein–Uhlenbeck operator on strictly convex domains in Gaussian space in terms of the Euclidean diameter D, by comparing to the gap of a corresponding one-dimensional Schrödinger model on (−D/2, D/2). The estimate confirms the Gaussian analogue of the fundamental gap conjecture and is shown to be sharp.

Because the setting is Gaussian, the authors note that measuring the domain’s size via the Euclidean diameter may be less natural than using a diameter computed with respect to a Gaussian metric. They explicitly pose the problem of whether their lower bound remains valid when the Euclidean diameter D is replaced by the Gaussian diameter D_G.