Inequalities for means of chords, with application to isoperimetric problems

Abstract

We consider a pair of isoperimetric problems arising in physics. The first concerns a Schrödinger operator in $L^2(\mathbb{R}^2)$ with an attractive interaction supported on a closed curve $Γ$, formally given by $-Δ-αδ(x-Γ)$; we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread $Γ$ in $\mathbb{R}^3$, homogeneously charged but not conducting, and we ask about the (renormalized) potential-energy minimizer. Both problems reduce to purely geometric questions about inequalities for mean values of chords of $Γ$. We prove an isoperimetric theorem for $p$-means of chords of curves when $p \leq 2$, which implies in particular that the global extrema for the physical problems are always attained when $Γ$ is a circle. The article finishes with a discussion of the $p$--means of chords when $p > 2$.