A NON LOCAL QUANTITATIVE CHARACTERIZATION OF ELLIPSES LEADING TO A SOLVABLE DIFFERENTIAL RELATION

@inproceedings{Amar2008ANL,
  title={A NON LOCAL QUANTITATIVE CHARACTERIZATION OF ELLIPSES LEADING TO A SOLVABLE DIFFERENTIAL RELATION},
  author={Makhlouf Amar and Lucio R. Berrone and Roberto Gianni},
  year={2008}
}
In this paper we prove that there are no domains E ⊂ R, other than the ellipses, such that the Lebesgue measure of the intersection of E and its homothetic image εE translated to a boundary point q ∈ ∂E is independent of q, provided thatE is "centered" at a certain interior pointO ∈ E (the center of homothety). Similar problems arise in various fields of mathematics, including, for example, the study of stationary isothermal surfaces and rearrangements.