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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Replacing convex sets by polytopes


Authors: S. Gallivan and J. Zaks
Journal: Proc. Amer. Math. Soc. 50 (1975), 351-357
MSC: Primary 52A20
MathSciNet review: 0380627
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Abstract: A conjecture of A. J. Hoffman is settled by showing that if $ P$ is a $ d$-polytope in $ {E^d}$, if compact convex subsets $ {C_1}, \cdots ,{C_k}$ are such that every $ t$-dimensional affine flat that meets $ P$ also meets $ \bigcup\nolimits_{i = 1}^k {{C_i}} $, then there exist polytopes $ {D_1}, \cdots ,{D_k}$, with $ {D_i} \subseteq {C_i}$ for all $ 1 \leqslant i \leqslant k$, such that every $ t$-flat that meets $ P$ also meets $ \bigcup\nolimits_{i = 1}^k {{D_i}} $, provided $ k \leqslant d - t + 1$. Counterexamples are given for the cases where $ d \leqslant 3,1 \leqslant t$ and $ k \geqslant d - t + 2$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0380627-1
PII: S 0002-9939(1975)0380627-1
Keywords: $ d$-polytope, affine flat, compact convex set
Article copyright: © Copyright 1975 American Mathematical Society