Abstract

Generalizing certain characteristic properties of ellipsoids, we describe all convex hypersurfaces in the n-dimensional real space, possibly unbounded, whose polar sets and point-source shadow-boundaries satisfy planarity conditions.

Author information

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

Valeriu Soltan

Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, U.S.A.

vsoltan@gmu.edu

Keywords

  • Body
  • convex
  • ellipsoid
  • polar set
  • quadric hypersurface
  • shadow-boundary
  • solid

Mathematics Subject Classification

52A20

References

  1. J. L. Arocha, L. Montejano, E. Morales: A quick proof of Höbinger-Burton-Larman’s theorem, Geom. Dedicata 63 (1996) 331–335.https://doi.org/10.1007/BF00181421Google Scholar
  2. G. R. Burton: Some characterisations of the ellipsoid, Israel J. Math. 28 (1977) 339–349.https://doi.org/10.1007/BF02760640Google Scholar
  3. P. M. Gruber: Über kennzeichnende Eigenschaften von euklidischen Räumen und Ellipsoiden. III, Monatsh. Math. 78 (1974) 311–340.https://doi.org/10.1007/BF01294643Google Scholar
  4. P. M. Gruber, T. Ódor: Ellipsoids are the most symmetric convex bodies, Arch. Math. 73 (1999) 394–400.https://doi.org/10.1007/s000130050414Google Scholar
  5. B. A. Ivanov: Straight line segments on the boundary of a convex body, Ukr. Geom. Sb. 13 (1973) 69–71.Google Scholar
  6. P. Kelly, E. G. Straus: On the projective centres of convex curves, Canad. J. Math. 12 (1960) 568-581.https://doi.org/10.4153/CJM-1960-050-7Google Scholar
  7. T. Kojima: On characteristic properties of the conic and quadric, Sci. Rep. Tôhoku Univ. 8 (1919) 67–78.Google Scholar
  8. T. Kubota: On the theory of closed convex surface, Proc. London Math. Soc. 14 (1914) 230–239.https://doi.org/10.1112/plms/s2_14.1.230Google Scholar
  9. T. Kubota: Einfache Beweise eines Satzes über die konvexe geschlossene Fläche, Sci. Rep. Tôhoku Univ. 3 (1914) 235–255.Google Scholar
  10. E. Makai, V. Soltan: Lower bounds on the numbers of shadow-boundaries and illuminated regions of a convex body, in: Intuitive Geometry (Szeged, 1991), K. Böröczky et al. (ed.), North-Holland, Amsterdam (1994) 249–268.Google Scholar
  11. A. Marchaud: Un théorème sur les corps convexes, Ann. Sci. Éc. Norm. Supér. 76 (1959) 283–304.https://doi.org/10.24033/asens.1084Google Scholar
  12. H. Martini, V. Soltan: Combinatorial problems on the illumination of convex bodies, Aequationes Math. 57 (1999) 121–152.https://doi.org/10.1007/s000100050074Google Scholar
  13. 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
  14. V. Soltan: Convex solids with planar homothetic sections through given points, J. Convex Analysis 16 (2009) 473–486.https://doi.org/10.68381/jca16025Google Scholar
  15. V. Soltan: Convex quadrics, Bul. Acad. Ştiinte Repub. Mold., Mat. 3 (2010) 94–106.Google Scholar
  16. V. Soltan: Convex solids with hyperplanar midsurfaces for restricted families of chords, Bul. Acad. Ştiinte Repub. Mold., Mat. 2 (2011) 23–40.Google Scholar
  17. V. Soltan: Convex solids whose point-source shadow-boundaries lie in hyperplanes, J. Geom. 103 (2012) 149–160.https://doi.org/10.1007/s00022-012-0114-6Google Scholar
  18. V. Soltan: Convex solids with hyperplanar shadow-boundaries, J. Convex Analysis 19 (2012) 591–607.https://doi.org/10.68381/jca19033Google Scholar
  19. V. Soltan: Convex quadrics and their characterizations by means of plane sections, in: Bridging Mathematics, Statistics, Engineering and Technology (Petersburg, 2011), B. Toni, et al. (ed.), Springer Proc. Math. Stat. 24, Springer, New York (2012) 131–145.https://doi.org/10.1007/978-1-4614-4559-3_12Google Scholar
  20. V. Soltan: Characterizations of convex quadrics in terms of plane quadric sections, midsurfaces, and shadow-boundaries, in: New Frontiers of Multidisciplinary Research in STEAM-H, B. Toni (ed), Springer Proc. Math. Stat. 90, Springer, New York (2014) 79–110.https://doi.org/10.1007/978-3-319-07755-0_4Google Scholar
  21. V. Soltan: Convex hypersurfaces with hyperplanar intersections of their homothetic copies, J. Convex Analysis 22 (2015) 145–159.https://doi.org/10.68381/jca22008Google Scholar
  22. V. Soltan: Lectures on Convex Sets, World Scientific, Hackensack (2015).https://doi.org/10.1142/9508Google Scholar

Suggested citation

V. Soltan. “Convex Surfaces with Planar Polar Sets and Point-Source Shadow-Boundaries.” Journal of Convex Analysis 24 (2017), No. 2, 645–660. https://doi.org/10.68381/jca24040

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.