  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}.
