We choose some special unit vectors n1,..., n5 in R3 and denote by L ⊂ R5 the set of all points (L1,..., L5) ∈ R5 with the following property: there exists a compact convex polytope P ⊂ R3 such that the vectors n1,..., n5 (and no other vector) are unit outward normals to the faces of P and the perimeter of the face with the outward normal nk is equal to Lk 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 (L1,..., L5) ∈ L, it is not possible to find a neighborhood U ⊂ R5 and an analytic set A ⊂ R5 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.

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