DOI: 10.68381/jca27064 ISSN: 0944-6532
Some Characterizations of the Ellipsoid by Centroids of Concurrent Sections
Zamantha Guerrero-Zarazua, Jesús Jerónimo-Castro, Francisco G. Jimenez-Lopez
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
\mathbb{R}^n
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 3-dimensional analog of this result for the case of centroids of concurrent sections. For every line
\ell
ℓ
intersecting a convex body in
\mathbb{R}^3
R
3
, we consider the set of centroids of the sections of K, cut by planes through
\ell
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.