## 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 language | English |
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Pages (from-to) | 247-254 |

Number of pages | 8 |

Journal | Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg |

Volume | 88 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Apr 2018 |

## Keywords

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