An Improved Bound for the Ovals Problem

Let $\gamma \subset {\mathbb{R}}^m,m\ge 2,$ be a closed curve of length $2\pi$ with its curvature $\kappa$, parametrized by arc length, and let ${\lambda }_{\gamma }$ be the first eigenvalue of the periodic curvature Schrödinger operator $-d^2\mathrm{/}d{s}^2+\kappa (s{)}^2$. We obtain $${\lambda }_{\gamma }\ge \frac{\sqrt{\pi }}{2}{\left(\frac{\mathrm{\Gamma }(7\mathrm{/}6)}{\mathrm{\Gamma }(5\mathrm{/}3)}\right)}^3\mathrm{.}$$ 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.