
Journal of Convex Analysis 27 (2020), No. 4, 12191231 Copyright Heldermann Verlag 2020 Some Characterizations of the Ellipsoid by Centroids of Concurrent Sections Zamantha GuerreroZarazua Facultad de Ingeniería, Universidad Autónoma de Querétaro, Querétaro, México Jesus JerónimoCastro Facultad de Ingeniería, Universidad Autónoma de Querétaro, Querétaro, México jesusjero@hotmail.com Francisco G. JimenezLopez Facultad de Ingeniería, Universidad Autónoma de Querétaro, Querétaro, México fjimenez@uaq.mx Recently M. Meyer and S. Reisner [Characterizations of ellipsoids by sectioncentroid location, Geometriae Dedicata 31 (1989) 345355] 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 R^{n} 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 3dimensional analog of this result for the case of centroids of concurrent sections. For every line L intersecting a convex body in R^{3}, 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. Keywords: Euclidean ball, ellipsoid, centroids. MSC: 52A15 [ Fulltextpdf (145 KB)] for subscribers only. 