Inequalities for means of chords, with application to isoperimetric problems

TL;DR

This paper establishes geometric inequalities for mean values of chords of curves, demonstrating that circles optimize certain physical properties related to Schrödinger operators and charged loops, with a focus on $p$-means for $p \,\leq\, 2$.

Contribution

It proves an isoperimetric theorem for $p$-means of chords of curves for $p \,\leq\, 2$, linking geometric inequalities to physical optimization problems.

Findings

Circles maximize the ground state energy in the Schrödinger operator problem.

Circles minimize potential energy for charged loops.

The theorem applies to $p$-means of chords when $p \,\leq\, 2$.

Abstract

We consider a pair of isoperimetric problems arising in physics. The first concerns a Schr\"odinger operator in $L^2(\mathbb{R}^2)$ with an attractive interaction supported on a closed curve $\Gamma$, formally given by $-\Delta-\alpha \delta(x-\Gamma)$; we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread $\Gamma$ 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 $\Gamma$. 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 $\Gamma$ is a circle. The article finishes with a discussion of the $p$--means of chords when $p > 2$.