Characterization of the circle by equipower properties (Q803942)

It is well-known that, if E is a disk, then every point of the plane not belonging to its boundary is a power point of E. The author considers the problem of finding how many distinct power points, interior or exterior to a convex body E, ensure that E is a disk. In the interior points case, she extends a result obtained by J. B. Kelly and J. Rosenbaum in the year 1946, proving that with the sole convexity assumption on E two interior power points are sufficient. In the exterior points case, she proves that three non-collinear power points ensure that E is a disk, whereas in general two or more collinear power points do not suffice. This result is obtained by using a measure theoretic argument.