A non local quantitative characterization of ellipses leading to a solvable differential relation

In this paper we prove that there are no domains, other than the ellipses, such that the Lebesgue measure of the intersection of such domains and their homothetic image translated to a boundary point is independent of such a point, provided that these domains are "centered" at a certain interior point (the center of homothety). Similar problems arise in various fields of mathematics, including, for example, the study of stationary isothermal surfaces and rearrangements.