Abstract

Extending results of C. A. Rogers ["Sections and projections of convex bodies", Portugal. Math. 24 (1965) 99–103], G. R. Burton ["Sections of convex bodies", J. London Math. Soc. 12 (1976) 331–336] and G. R. Burton and P. Mani ["A characterization of the ellipsoid in terms of concurrent sections, Comment. Math. Helv. 53 (1978) 485–507] to the case of unbounded convex sets, we prove that line-free closed convex sets K1K_1 and K2K_2 of dimension nn in Rn\R^n, n≥4n \ge 4, are homothetic provided there are points p1∈int K1p_1 \in \Int K_1 and p2∈int K2p_2 \in \Int K_2 such that for every pair of parallel 2-dimensional planes L1L_1 and L2L_2 through p1p_1 and p2p_2, respectively, the sections K1∩L1K_1 \cap L_1 and K2∩L2K_2 \cap L_2 are homothetic. Furthermore, if there is a homothety f:Rn→Rnf: \R^n \to \R^n such that f(K1)=K2f(K_1) = K_2 and f(p1)≠p2f(p_1) \ne p_2, then K1K_1 and K2K_2 are convex cones or their boundaries are convex quadric surfaces. Related results on elliptic and centrally symmetric 2-dimensional bounded sections of convex sets are considered.

Author information

Contact details are reproduced from the original publication and may be historical.

Val Soltan

Dept. of Mathematical Sciences, George Mason University, Fairfax, VA 22030, U.S.A.

vsoltan@gmu.edu

Keywords

  • Homothety
  • convex body
  • planar section
  • quadric surface

Mathematics Subject Classification

52A20

References

  1. P. W. Aitchison, C. M. Petty, C. A. Rogers: A convex body with a false centre is an ellipsoid, Mathematika, Lond. 18 (1971) 50–59.https://doi.org/10.1112/S0025579300008366Google Scholar
  2. T. Bonnesen, W. Fenchel: Theorie der konvexen Körper, Springer, Berlin (1934); Engl. transl.: Theory of Convex Bodies, BCS, Moscow, USA (1987).https://doi.org/10.1007/978-3-642-47404-0Google Scholar
  3. G. R. Burton: Sections of convex bodies, J. Lond. Math. Soc., II. Ser. 12 (1976) 331–336.https://doi.org/10.1112/jlms/s2-12.3.331Google Scholar
  4. G. R. Burton, P. Mani: A characterization of the ellipsoid in terms of concurrent sections, Comment. Math. Helv. 53 (1978) 485–507.https://doi.org/10.1007/BF02566093Google Scholar
  5. H. Busemann: The Geometry of Geodesics, Academic Press, New York (1955).Google Scholar
  6. M. Florenzano, C. Le Van: Finite Dimensional Convexity and Optimization, Springer, Berlin (2001).https://doi.org/10.1007/978-3-642-56522-9Google Scholar
  7. R. J. Gardner: Geometric Tomography, Cambridge University Press, Cambridge (1995).Google Scholar
  8. J.-B. Hiriart-Urruty, C. Lemaréchal: Fundamentals of Convex Analysis, Springer, Berlin (2001).https://doi.org/10.1007/978-3-642-56468-0Google Scholar
  9. J. Höbinger: Über einen Satz von Aitchison, Petty und Rogers, Ph. D. Thesis, Technical University Wien, Wien (1974).Google Scholar
  10. V. L. Klee: Extremal structure of convex sets, Arch. Math. 8 (1957) 234–240.https://doi.org/10.1007/BF01899998Google Scholar
  11. T. Kubota: Einfache Beweise eines Satzes über die konvexe geschlossene Fläche, Tôhoku Science Rep. 3 (1914) 235–255.Google Scholar
  12. D. G. Larman: A note on the false centre problem, Mathematika, Lond. 21 (1974) 216–227.https://doi.org/10.1112/S0025579300008603Google Scholar
  13. D. G. Larman, E. Morales-Amaya: On the false pole problem, Monatsh. Math. 151 (2007) 271–286.https://doi.org/10.1007/s00605-007-0448-6Google Scholar
  14. M. A. Penna, R. R. Patterson: Projective Geometry and its Applications to Computer Graphics, Prentice-Hall, Englewood Cliffs (1986).Google Scholar
  15. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  16. C. A. Rogers: Sections and projections of convex bodies, Port. Math. 24 (1965) 99–103.Google Scholar
  17. G. E. Shilov: Linear Algebra, Dover, New York (1977).Google Scholar
  18. V. Soltan: Sections and projections of homothetic convex bodies, J. Geom. 84 (2005) 152–163.https://doi.org/10.1007/s00022-005-0026-9Google Scholar
  19. V. Soltan: Convex solids with planar midsurfaces, Proc. Amer. Math. Soc. 136 (2008) 1071–1081.https://doi.org/10.1090/S0002-9939-07-09125-3Google Scholar

Suggested citation

V. Soltan. “Convex Solids with Planar Homothetic Sections Through Given Points.” Journal of Convex Analysis 16 (2009), No. 2, 473–486. https://doi.org/10.68381/jca16025

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.