Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Convex solids with planar midsurfaces


Author: Valeriu Soltan
Journal: Proc. Amer. Math. Soc. 136 (2008), 1071-1081
MSC (2000): Primary 52A20
Published electronically: November 30, 2007
MathSciNet review: 2361883
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the boundary of an $ n$-dimensional closed convex set $ B \subset \mathbb{R}^n$, possibly unbounded, is a convex quadric surface if and only if the middle points of every family of parallel chords of $ B$ lie in a hyperplane. To prove this statement, we show that the boundary of $ B$ is a convex quadric surface if and only if there is a point $ p \in \mathrm{int}\,B$ such that all sections of $ \mathrm{bd}\,B$ by 2-dimensional planes through $ p$ are convex quadric curves. Generalizations of these statements that involve boundedly polyhedral sets are given.


References [Enhancements On Off] (What's this?)

  • 1. W.Blaschke, Kreis und Kugel, Viet, Leipzig, 1916.
  • 2. H.Brunn, Ueber Kurven ohne Wendepunkte, Habilitationschrift, T.Ackermann, München, 1889.
  • 3. Herbert Busemann, The geometry of geodesics, Academic Press Inc., New York, N. Y., 1955. MR 0075623 (17,779a)
  • 4. Peter Gruber, Über kennzeichnende Eigenschaften von euklidischen Räumen und Ellipsoiden. I, J. Reine Angew. Math. 265 (1974), 61–83 (German). MR 0338931 (49 #3694)
  • 5. V. L. Klee Jr., Extremal structure of convex sets, Arch. Math. (Basel) 8 (1957), 234–240. MR 0092112 (19,1065a)
  • 6. Victor Klee, Some characterizations of convex polyhedra, Acta Math. 102 (1959), 79–107. MR 0105651 (21 #4390)
  • 7. T.Kubota, Einfache Beweise eines Satzes über die konvexe geschlossene Fläche, Sci. Rep. Tôhoku Univ. 3 (1914), 235-255.
  • 8. T.Kubota, On a characteristic property of the ellipse, Tôhoku Math. J. 9 (1916), 148-151.
  • 9. M.A.Penna, R.R.Patterson, Projective geometry and its applications to computer graphics, Prentice-Hall, NJ, 1986.
  • 10. Georgi E. Shilov, Linear algebra, Revised English edition, Dover Publications Inc., New York, 1977. Translated from the Russian and edited by Richard A. Silverman. MR 0466162 (57 #6043)
  • 11. V.Snyder, C.H.Sisam, Analytic geometry of space, Holt and Co., New York, 1937.
  • 12. Valeriu Soltan, Convex bodies with polyhedral midhypersurfaces, Arch. Math. (Basel) 65 (1995), no. 4, 336–341. MR 1349188 (96h:52004), http://dx.doi.org/10.1007/BF01195545
  • 13. Roger Webster, Convexity, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1994. MR 1443208 (98h:52001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 52A20

Retrieve articles in all journals with MSC (2000): 52A20


Additional Information

Valeriu Soltan
Affiliation: Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, Virginia 22030
Email: vsoltan@gmu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09125-3
PII: S 0002-9939(07)09125-3
Keywords: Convex quadric surface, midsurface, planar section, polyhedron.
Received by editor(s): December 8, 2006
Published electronically: November 30, 2007
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2007 American Mathematical Society