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Inequalities for means of chords, with application to isoperimetric problems

Pavel Exner Evans M Harrell Michael Loss

TBD mathscidoc:1910.43073

Letters in Mathematical Physics, 75, (3), 225-233, 2006.3
We consider a pair of isoperimetric problems arising in physics. The first concerns a Schrdinger operator in L^2(\mathbb{R}^2) with an attractive interaction supported on a closed curve , formally given by (<i>x</i>); we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread in L^2(\mathbb{R}^2), 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 . We prove an isoperimetric theorem for <i>p</i>-means of chords of curves when <i>p</i> 2, which implies in particular that the global extrema for the physical problems are always attained when is a circle. The letter concludes with a discussion of the <i>p</i>-means of chords when <i>p</i> &gt; 2.
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@inproceedings{pavel2006inequalities,
  title={Inequalities for means of chords, with application to isoperimetric problems},
  author={Pavel Exner, Evans M Harrell, and Michael Loss},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020125014138657602},
  booktitle={Letters in Mathematical Physics},
  volume={75},
  number={3},
  pages={225-233},
  year={2006},
}
Pavel Exner, Evans M Harrell, and Michael Loss. Inequalities for means of chords, with application to isoperimetric problems. 2006. Vol. 75. In Letters in Mathematical Physics. pp.225-233. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020125014138657602.
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