Convex solids with planar midsurfaces

Soltan, Valeriu

doi:10.1090/S0002-9939-07-09125-3

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.

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