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On a conjecture of A. J. Hoffman


Author: Joseph Zaks
Journal: Proc. Amer. Math. Soc. 27 (1971), 122-125
MSC: Primary 52.10
DOI: https://doi.org/10.1090/S0002-9939-1971-0275282-1
MathSciNet review: 0275282
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Abstract: A $ 3$-polytope $ P$ and four closed convex sets $ {C_1}, \cdots ,{C_4}$ in $ P$ are described, having the following property: every line which meets $ P$ meets at least one of the $ {C_i}$'s, and for every collection of polytopes $ {D_1}, \cdots ,{D_4}$, with $ {D_i} \subseteq {C_i}$ for all $ 1 \leqq i \leqq 4$, there exists a line which meets $ P$ and misses all of the $ {D_i}$'s. This is a counterexample to a conjecture of A. J. Hoffman.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0275282-1
Keywords: Convex $ d$-polytype, closed convex set, affine $ t$-flat, convex planar curve
Article copyright: © Copyright 1971 American Mathematical Society

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