Benguria–Loss ovals conjecture

Establish that C_oval = 1, where C_oval is the infimum of the lowest eigenvalue λ_0(γ) of the Schrödinger operator H_γ = −d^2/ds^2 + κ(s)^2 over all simple closed convex plane curves γ of length 2π parameterized by arc length.

Background

The quantity C_oval arises in connections with spectral inequalities (e.g., Lieb–Thirring-type bounds) and geometric analysis of plane curves. The unit circle yields the upper bound 1, and nontrivial lower bounds have been established, but the exact value is unknown.

A sharp characterization would resolve a central question linking curvature-dependent Schrödinger operators to isoperimetric-type extremal problems on plane ovals.