Some Characterizations of the Ellipsoid by Centroids of Concurrent Sections

Recently M. Meyer and S. Reisner [Characterizations of ellipsoids by section-centroid location, Geometriae Dedicata 31 (1989) 345--355] proved the following result, generalizing a classical result due to Brunn [see W. Blaschke, Kreis und Kugel, Göschen Verlag, Leipzig (1916)]: If the subset K of Rn is a convex body with the property that the centroids of every set of parallel sections, cut by parallel hyperplanes, are collinear, then K is an ellipsoid. In this paper we analyze the 3-dimensional analog of this result for the case of centroids of concurrent sections. For every line L intersecting a convex body in R3, we consider the set of centroids of the sections of K, cut by planes through L, and we assume the locus of these centroids determine a planar, differentiable simple closed curve with no segments. In such a case we prove that K is an ellipsoid. Furthermore, we prove that if for a convex body K there are two parallel planes such that for every line in any of these planes the associated locus of centroids is a circle, then K is a Euclidean ball.