Inequalities for means of chords, with application to isoperimetric problems

Details Inequalities for means of chords, with application to isoperimetric problems Inequalities for means of chords, with application to isoperimetric problems

2005-08-31

arxiv.org/abs/math-ph/0508060v1

Lett. Math. Phys. 75 (2006), 225-233 ; addendum 75 (2006), 219

Physics

Mathematical Physics

LaTeX2e, 11 pages

Scientific paper

10.1007/s11005-006-0053-y

We consider a pair of isoperimetric problems arising in physics. The first concerns a Schr\"odinger operator in $L^2(\mathbb{R}^2)$ with an attractive interaction supported on a closed curve $\Gamma$, formally given by $-\Delta-\alpha \delta(x-\Gamma)$; we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread $\Gamma$ in $\mathbb{R}^3$, homogeneously charged but not conducting, and we ask about the (renormalized) potential-energy minimizer. Both problems reduce to purely geometric questions about inequalities for mean values of chords of $\Gamma$. We prove an isoperimetric theorem for $p$-means of chords of curves when $p \leq 2$, which implies in particular that the global extrema for the physical problems are always attained when $\Gamma$ is a circle. The article finishes with a discussion of the $p$--means of chords when $p > 2$.

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