On the critical exponent in an isoperimetric inequality for chords

TL;DR

This paper investigates the maximization of $L^p$ norms of chords on closed curves, identifying a critical exponent where the circle ceases to be a local maximizer, correcting previous misconceptions.

Contribution

It determines the critical value $p_c(u)$ for the isoperimetric inequality of chords, clarifying when the circle is no longer optimal.

Findings

The circle is the unique maximizer for $1 \,\leq\, p \leq 2$.

The critical exponent $p_c(\frac{1}{2}L)=\frac{5}{2}$ is identified.

Previous claims about the maximizer shape are corrected.

Abstract

The problem of maximizing the $L^p$ norms of chords connecting points on a closed curve separated by arclength $u$ arises in electrostatic and quantum--mechanical problems. It is known that among all closed curves of fixed length, the unique maximizing shape is the circle for $1 \le p \le 2$, but this is not the case for sufficiently large values of $p$. Here we determine the critical value $p_c(u)$ of $p$ above which the circle is not a local maximizer finding, in particular, that $p_c(\frac12 L)=\frac52$. This corrects a claim made in \cite{EHL}.