On the critical exponent in an isoperimetric inequality for chords

Details On the critical exponent in an isoperimetric inequality for chords On the critical exponent in an isoperimetric inequality for chords

2007-03-05

arxiv.org/abs/math-ph/0703020v1

Physics

Mathematical Physics

LaTeX, 12 pages, with 1 eps figure

Scientific paper

10.1016/j.physleta.2007.03.067

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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