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On hyperplanes and polytopes

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Abstract

We call a convex subsetN of a convexd-polytopeP⊂E d ak-nucleus ofP ifN meets everyk-face ofP, where 0<k<d. We note thatP has disjointk-nuclei if and only if there exists a hyperplane inE d which bisects the (relative) interior of everyk-face ofP, and that this is possible only if [d+2/2]≤k≤d−1. Our main results are that any convexd-polytope with at most 2d−1 vertices (d≥3) possesses disjoint (d−1)-nuclei and that 2d−1 is the largest possible number with this property. Furthermore, every convexd-polytope with at most 2d facets (d≥3) possesses disjoint (d−1)-nuclei, 2d cannot be replaced by 2d+2, and ford=3, six cannot be replaced by seven.

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Authors and Affiliations

  1. Dept. of Geometry, Eötvös L. University, Rákóczi út 5, 1088, Budapest, Hungary

    K. Bezdek

  2. Dept. of Mathematics, Cornell University, 14853, Ithaca, NY, U.S.A.

    K. Bezdek & R. Connelly

  3. Dept. of Mathematics & Statistics, University of Calgary, T2N 1N4, Calgary, Alberta, Canada

    T. Bisztriczky

Authors
  1. K. Bezdek
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  2. T. Bisztriczky
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  3. R. Connelly
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Additional information

Partially supported by Hung. Nat. Found. for Sci. Research number 1238.

Partially supported by the Natural Sciences and Engineering Council of Canada.

Partially supported by N.S.F. grant number MCS-790251.

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Bezdek, K., Bisztriczky, T. & Connelly, R. On hyperplanes and polytopes. Monatshefte für Mathematik 109, 39–48 (1990). https://doi.org/10.1007/BF01298851

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  • DOI: https://doi.org/10.1007/BF01298851

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