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On the critical exponent in an isoperimetric inequality for chords

Pavel Exner·Martin Fraas·Evans M. Harrell II
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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}.

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Pavel Exner, Martin Fraas, Evans M. Harrell II. 2007-03-05. On the critical exponent in an isoperimetric inequality for chords. https://doi.org/10.1016/j.physleta.2007.03.067

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