Equichordal Problem
Is there a planar body bounded by a simple closed curve and star-shaped with respect to two interior points 
and 
whose point X-rays at and are both constant? Rychlik (1997) has answered the question in the negative.
References
Rychlik, M. ``The Equichordal Point Problem.'' Elec. Res. Announcements Amer. Math. Soc. 2, 108-123, 1996. Rychlik, M. ``A Complete Solution to the Equichordal Problem of Fujiwara, Blaschke, Rothe, and Weitzenböck.'' Invent. Math. 129, 141-212, 1997.
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