Replacing convex sets by polytopes
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- by S. Gallivan and J. Zaks PDF
- Proc. Amer. Math. Soc. 50 (1975), 351-357 Request permission
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 $k \geqslant d - t + 2$.References
- H. G. Eggleston, Intersection of convex sets, J. London Math. Soc. (2) 5 (1972), 753–754. MR 313941, DOI 10.1112/jlms/s2-5.4.753 B. Grünbaum, Convex polytopes, Pure and Appl. Math., vol. 16, Interscience, New York, 1967. MR 37 # 2085.
- W. R. Hare Jr. and C. R. Smith, On convex subsets of a polytope, Proc. Amer. Math. Soc. 35 (1972), 238–239. MR 301633, DOI 10.1090/S0002-9939-1972-0301633-6
- A. J. Hoffman, On the covering of polyhedra by polyhedra, Proc. Amer. Math. Soc. 23 (1969), 123–126. MR 247570, DOI 10.1090/S0002-9939-1969-0247570-7
- Victor Klee, Some characterizations of convex polyhedra, Acta Math. 102 (1959), 79–107. MR 105651, DOI 10.1007/BF02559569
- Joseph Zaks, On a conjecture of A. J. Hoffman, Proc. Amer. Math. Soc. 27 (1971), 122–125. MR 275282, DOI 10.1090/S0002-9939-1971-0275282-1
- Joseph Zaks, On a conjecture of A. J. Hoffman. II, Proc. Amer. Math. Soc. 34 (1972), 215–221. MR 296810, DOI 10.1090/S0002-9939-1972-0296810-7
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 50 (1975), 351-357
- MSC: Primary 52A20
- DOI: https://doi.org/10.1090/S0002-9939-1975-0380627-1
- MathSciNet review: 0380627