The problem of maximizing the L p norms of chords connecting points on a closed curve separated by arc length 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 p 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 (1 2 L)= 5 2. This corrects a claim made in [P. Exner, EM Harrell, M. Loss, Lett. Math. Phys. 75 (2006) 225].