Convex solids with planar midsurfaces
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- by Valeriu Soltan PDF
- Proc. Amer. Math. Soc. 136 (2008), 1071-1081 Request permission
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
- W. Blaschke, Kreis und Kugel, Viet, Leipzig, 1916.
- H. Brunn, Ueber Kurven ohne Wendepunkte, Habilitationschrift, T. Ackermann, München, 1889.
- Herbert Busemann, The geometry of geodesics, Academic Press, Inc., New York, N.Y., 1955. MR 0075623
- Peter Gruber, Über kennzeichnende Eigenschaften von euklidischen Räumen und Ellipsoiden. I, J. Reine Angew. Math. 265 (1974), 61–83 (German). MR 338931, DOI 10.1515/crll.1974.265.61
- V. L. Klee Jr., Extremal structure of convex sets, Arch. Math. (Basel) 8 (1957), 234–240. MR 92112, DOI 10.1007/BF01899998
- Victor Klee, Some characterizations of convex polyhedra, Acta Math. 102 (1959), 79–107. MR 105651, DOI 10.1007/BF02559569
- T. Kubota, Einfache Beweise eines Satzes über die konvexe geschlossene Fläche, Sci. Rep. Tôhoku Univ. 3 (1914), 235–255.
- T. Kubota, On a characteristic property of the ellipse, Tôhoku Math. J. 9 (1916), 148–151.
- M. A. Penna, R. R. Patterson, Projective geometry and its applications to computer graphics, Prentice-Hall, NJ, 1986.
- 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
- V. Snyder, C. H. Sisam, Analytic geometry of space, Holt and Co., New York, 1937.
- Valeriu Soltan, Convex bodies with polyhedral midhypersurfaces, Arch. Math. (Basel) 65 (1995), no. 4, 336–341. MR 1349188, DOI 10.1007/BF01195545
- Roger Webster, Convexity, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR 1443208
Additional Information
- Valeriu Soltan
- Affiliation: Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, Virginia 22030
- Email: vsoltan@gmu.edu
- Received by editor(s): December 8, 2006
- Published electronically: November 30, 2007
- Communicated by: Jon G. Wolfson
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 1071-1081
- MSC (2000): Primary 52A20
- DOI: https://doi.org/10.1090/S0002-9939-07-09125-3
- MathSciNet review: 2361883