An isoperimetric problem for leaky loops and related mean-chord inequalities

Abstract

We consider a class of Hamiltonians in $L^2(\R^2)$ with attractive interaction supported by piecewise $C^2$ smooth loops $Γ$ of a fixed length $L$, formally given by $-Δ-αδ(x-Γ)$ with $α>0$. It is shown that the ground state of this operator is locally maximized by a circular $Γ$. We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of $Γ$.