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Discrete & Computational Geometry

, Volume 34, Issue 1, pp 25–45 | Cite as

Partitions of a Polytope and Mappings of a Point Set to Facets

  • Roman N. Karasev
Article

Abstract

Three theorems of this paper generalize previous results of the author on conjectures of A. Bezdek and V.V. Proizvolov. They show the existence of mappings from a given point set to the set of facets of a given polytope that satistfy some special conditions. Developing the same technique, some results on convex polytope partitions are presented, two of them dealing with partitions with prescribed measures of parts. Then we prove a corollary on the existence of a possibly nonconvex polytope with a given set of vertices, containing given points in its interior. We also consider problems of the following type: find an assignment of vectors from a given set to the parts of a given convex partition of ℝn so that the shifts of the parts by their corresponding vectors either do not intersect by interior points or cover ℝn

Keywords

Computational Mathematic Interior Point Convex Partition Prescribe Measure Nonconvex Polytope 
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Copyright information

© Springer 2005

Authors and Affiliations

  • Roman N. Karasev
    • 1
  1. 1.Department of Mathematics, Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, 141700Russia

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