Quantum Lunch: On the oval conjecture

Speaker: Jérémy Sok from QMATH

On the oval conjecture

In connection with a Lieb-Thirring inequality in dimension 1, Benguria and Loss derived an isoperimetric problem for planar curves, the oval conjecture. Given a twice-differentiable simple closed curve of length 2π in the plane, let κ(s) be its curvature written as a function of the arc-length parameter. The oval conjecture states that the ground state energy of -Δ+κ(s)2 on the circle is bounded from below by 1 and that the infimum is attained by a family ovals containing the unit circle. We will review the partial results obtained so far and discuss different aspects of the problem.