On the critical exponent in an isoperimetric inequality for chords

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 $p_c(\frac{1}{2}L)=\frac{5}{2}$. This corrects a claim made in \cite{EHL}.

LaTeX, 12 pages, with 1 eps figure

References in corpus (3)

• Circles Minimize most Knot Energies
• An isoperimetric problem for leaky loops and related mean-chord inequalities
• Inequalities for means of chords, with application to isoperimetric problems

Cited by in corpus (4)

• A spectral isoperimetric inequality for cones
• On geometric perturbations of critical Schrödinger operators with a surface interaction
• Pareto optimization of resonances and minimum-time control
• On the maximization of a class of functionals on convex regions, and the characterization of the farthest convex set