Pavel Exner, Martin Fraas, Evans M. Harrell II
On the critical exponent in an isoperimetric inequality for chords
(209K, pdf)

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}.