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A Non Local Quantitative Characterization of ellipses Leading to a Solvable Differential Relation  
 
  Authors: M. Amar, L.R. Berrone, R. Gianni,  
  Keywords: Convex sets, asymptotic expansion, ordinary differential equations.  
  Date Received: 14/02/08  
  Date Accepted: 03/07/08  
  Subject Codes:

52A10, 41A58, 34A05.

  
  Editors: Catherine Bandle,  
 
  Abstract:

In this paper we prove that there are no domains $ \mathcal{E} \subset \mathbb{R}^{2}$, other than the ellipses, such that the Lebesgue measure of the intersection of $ \mathcal{E}$ and its homothetic image $ \varepsilon \mathcal{E}$ translated to a boundary point $ q\in \partial \mathcal{E}$ is independent of $ q$, provided that $ \mathcal{E}$ is "centered" at a certain interior point $ O\in \mathcal{E}$ (the center of homothety).
Similar problems arise in various fields of mathematics, including, for example, the study of stationary isothermal surfaces and rearrangements. ;


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