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.

References

It was conjectured in that the upper bound was in fact sharp, thus C_{\ref{oval}=1.

— Mathematical exploration and discovery at scale  (2511.02864 - Georgiev et al., 3 Nov 2025) in Subsection “The Ovals problem” (Section 4.9)

The Ovals conjecture of Benguria and Loss asserts that $1$ is also a universal lower bound.

— An Improved Bound for the Ovals Problem  (2609.10775 - Suragan, 9 Sep 2026) in Section 1, Introduction and main results

To the best of our knowledge, the sharp lower bound in eq:C-oval-def therefore remains open.

eq:C-oval-def:

Coval:=inf⁡γ∈Cλγ.C_{\mathrm{oval}}:=\inf_{\gamma\in\mathfrak C}\lambda_\gamma.

— An Improved Bound for the Ovals Problem  (2609.10775 - Suragan, 9 Sep 2026) in Section 1, Introduction and main results

Benguria and Loss conjectured that $C=1$ and exhibited a continuous equality family containing the circle and noncircular ovals~\citep{bengurialoss2004,burchardthomas2005,bernsteinmettler2015}, proving $C\leq1$, while Linde proved the global lower bound $C>0.81$; numerical evaluation of the explicit constant in his theorem gives $C>0.8246$~\citep{linde2025}.

— Autonomous Mathematical Discovery in an Open-World Multi-Agent Environment  (2608.23691 - Chung et al., 24 Aug 2026) in Section 3, subsection “Ovals problem”