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On convex subsets of a polytope


Authors: W. R. Hare and C. R. Smith
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
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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$.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0301633-6
Keywords: Convex polytope, closed convex set, affine flat, convex hull
Article copyright: © Copyright 1972 American Mathematical Society

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