Abstract

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\mathbb{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 ℓ\ell intersecting a convex body in R3\mathbb{R}^3, we consider the set of centroids of the sections of K, cut by planes through ℓ\ell, 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.

Author information

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Zamantha Guerrero-Zarazua

Facultad de Ingeniería, Universidad Autónoma de Querétaro, Querétaro, México

Jesús Jerónimo-Castro

Facultad de Ingeniería, Universidad Autónoma de Querétaro, Querétaro, México

jesusjero@hotmail.com

Francisco G. Jimenez-Lopez

Facultad de Ingeniería, Universidad Autónoma de Querétaro, Querétaro, México

fjimenez@uaq.mx

Keywords

  • Euclidean ball
  • ellipsoid
  • centroids

Mathematics Subject Classification

52A15

References

  1. Archimedes: De corporibus fluitantibus, Opera Omnia 2 (1913) 317–413.Google Scholar
  2. W. Blaschke: Kreis und Kugel, Göschen, Leipzig (1916).https://doi.org/10.1515/9783112392348Google Scholar
  3. W. Blaschke: Über affine Geometrie IX: Verschiedene Bemerkungen und Aufgaben, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 69 (1917) 412–420.Google Scholar
  4. K. Borsuk: Drei Sätze über die n-dimensionale Sphäre, Fund. Math. 20 (1933) 177–190.https://doi.org/10.4064/fm-20-1-177-190Google Scholar
  5. H. Brunn: Über Ovale und Eiflächen, Dissertation, München (1887).Google Scholar
  6. G. R. Burton, P. Mani: A characterisation of the ellipsoid in terms of concurrent sections, Comment. Math. Helvetici 53 (1978) 485–507.https://doi.org/10.1007/BF02566093Google Scholar
  7. H. Flanders: A further comment on Pappus, Amer. Math. Monthly 77 (1970) 965–968.https://doi.org/10.1080/00029890.1970.11992639Google Scholar
  8. H. Groemer: Geometric Applications of Fourier Series and Spherical Harmonics, Cambridge University Press, Cambridge (1996).https://doi.org/10.1017/CBO9780511530005Google Scholar
  9. B. Grünbaum: Measures of symmetry for convex sets, Proc. Sympos. Pure Math., Vol. VII (1963) 233–270.https://doi.org/10.1090/pspum/007/0156259Google Scholar
  10. J. Jerónimo-Castro: Further characterization of ellipsoids, Monatshefte Math. 160 (2009) 73–80.https://doi.org/10.1007/s00605-008-0091-xGoogle Scholar
  11. M. Meyer, S. Reisner: Characterizations of ellipsoids by section-centroid location, Geometriae Dedicata 31 (1989) 345–355.https://doi.org/10.1007/BF00147465Google Scholar

Suggested citation

Z. Guerrero-Zarazua, J. Jerónimo-Castro, F. G. Jimenez-Lopez. “Some Characterizations of the Ellipsoid by Centroids of Concurrent Sections.” Journal of Convex Analysis 27 (2020), No. 4, 1219–1231. https://doi.org/10.68381/jca27064

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press.