A convex body in RP^n is an ellipsoid if its hypersections through two fixed interior points are paired by harmonic homologies for every (n-2)-plane in a given hyperplane.
On the false pole problem.Mh Math151, 271-286 (2007)
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A characterization of the ellipsoid in terms of pairs of sections associated by a harmonic homology
A convex body in RP^n is an ellipsoid if its hypersections through two fixed interior points are paired by harmonic homologies for every (n-2)-plane in a given hyperplane.