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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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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Keywords: Convex <IMG WIDTH="16" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$d$">-polytype, closed convex set, affine <IMG WIDTH="13" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$t$">-flat, convex planar curve
Article copyright: © Copyright 1971 American Mathematical Society