Characterization of the circle by equipower properties

Equipower and Equichordal Extension

“…A counterexample was already given by K.Yanagihara in [10], [11]. Further, L. Zuccheri in [12] proved that any convex region with two interior power points is a disk, without further assumption on its boundary, and extended the question to exterior points.…”

A list of problems in Plane Geometry with simple statement that remain unsolved

“…This problem could have been included in the previous family, but it deserves to be highlighted. Show that the circle is the unique C ∞ Jordan curve with an equipower point (see [41] for the definition). It is easy to prove that lunes actually have an equipower point, but they are only piecewise C ∞ .…”