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

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.