Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?

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Abstract

We choose some special unit vectors n 1, … , n 5 in R 3 and denote by L⊂ R 5 the set of all points (L 1, … , L 5) ∈ R 5 with the following property: there exists a compact convex polytope P⊂ R 3 such that the vectors n 1, … , n 5 (and no other vector) are unit outward normals to the faces of P and the perimeter of the face with the outward normal n k is equal to L k for all k= 1 , … , 5. Our main result reads that L is not a locally-analytic set, i.e., we prove that, for some point (L 1, … , L 5) ∈ L, it is not possible to find a neighborhood U⊂ R 5 and an analytic set A⊂ R 5 such that L∩ U= A∩ U. We interpret this result as an obstacle for finding an existence theorem for a compact convex polytope with prescribed directions and perimeters of the faces.

Original languageEnglish
Pages (from-to)247-254
Number of pages8
JournalAbhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
Volume88
Issue number1
DOIs
Publication statusPublished - 1 Apr 2018

Keywords

  • Analytic set
  • Convex polyhedron
  • Euclidean space
  • Perimeter of a face

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