On the critical exponent in an isoperimetric inequality for chords

The problem of maximizing the Lp 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 pc(u) of p above which the circle is not a local maximizer finding, in particular, that pc(12L)=52. This corrects a claim made in [P. Exner, E.M. Harrell, M. Loss, Lett. Math. Phys. 75 (2006) 225].