On convex subsets of a polytope
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- by W. R. Hare and C. R. Smith PDF
- Proc. Amer. Math. Soc. 35 (1972), 238-239 Request permission
Abstract:
A. J. Hoffman conjectured the following: Given a d-polytope P and a collection, ${C_1}, \cdots ,{C_k}$, of closed convex subsets of P with the property that each t-flat, $0 \leqq t \leqq d - 1$, which meets P also meets some ${C_i}$, then there exist polytopes ${D_j} \subset {C_j}$ such that every t-flat which meets P also meets some ${D_j}$. In this note it is shown that the above is true for $k = 2$.References
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B. Grünbaum, Convex polytopes, Pure and Appl. Math., vol. 16, Interscience, New York, 1967. MR 37 #2085.
- 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
- 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 —, On Hoffman’s conjecture (manuscript).
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 238-239
- MSC: Primary 52A25
- DOI: https://doi.org/10.1090/S0002-9939-1972-0301633-6
- MathSciNet review: 0301633