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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
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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.

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Additional Information

Valeriu Soltan
Affiliation: Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, Virginia 22030

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

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