An Improved Bound for the Ovals Problem

Abstract

Let $γ\subset\mathbb R^{m},\,m\geq2,$ be a closed curve of length $2π$ with its curvature $κ$, parametrized by arc length, and let $λ_γ$ be the first eigenvalue of the periodic curvature Schrödinger operator $-d^2/d s^2+κ(s)^2$. We obtain \[ λ_γ\geq \frac{\sqrtπ}{2} \left(\frac{Γ(7/6)}{Γ(5/3)}\right)^3. \] This is a near-sharp lower bound for the Ovals problem. Our proof introduces a new geometric approach. We derive a convolution identity from the closure condition and combine it with projection averaging over tangent directions and sharp Poincaré inequalities on antipodal arcs. As applications, we provide an improved two-state kinetic Lieb-Thirring inequality and the corresponding two-eigenvalue constant.