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

We consider a class of Hamiltonians in L2(R2) with attractive interaction supported by piecewise C2 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. [ABSTRACT FROM AUTHOR]