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Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces? (English) Zbl 1392.52007
Summary: We choose some special unit vectors \({\mathbf {n}}_1,\dots ,{\mathbf {n}}_5\) in \({\mathbb {R}}^3\) and denote by \({\mathcal {L}}\subset {\mathbb {R}}^5\) the set of all points \((L_1,\dots ,L_5)\in {\mathbb {R}}^5\) with the following property: there exists a compact convex polytope \(P\subset {\mathbb {R}}^3\) such that the vectors \({\mathbf {n}}_1,\dots ,{\mathbf {n}}_5\) (and no other vector) are unit outward normals to the faces of \(P\) and the perimeter of the face with the outward normal \({\mathbf {n}}_k\) is equal to \(L_k\) for all \(k=1,\dots ,5\). Our main result reads that \({\mathcal {L}}\) is not a locally-analytic set, i.e., we prove that, for some point \((L_1,\dots ,L_5)\in {\mathcal {L}}\), it is not possible to find a neighborhood \(U\subset {\mathbb {R}}^5\) and an analytic set \(A\subset {\mathbb {R}}^5\) such that \({\mathcal {L}}\cap U=A\cap U\). 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.

MSC:
52B10 Three-dimensional polytopes
51M20 Polyhedra and polytopes; regular figures, division of spaces
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[1] Alexandrov, A.D.: An elementary proof of the Minkowski and some other theorems on convex polyhedra. Izv. Akad. Nauk SSSR, Ser. Mat. (4), 597-606 (1937) (in Russian)
[2] Alexandrov, A.D.: Selected Works. Part 1: Selected Scientific Papers. Gordon and Breach Publishers, Amsterdam (1996) · Zbl 0960.01035
[3] Alexandrov, A.D.: Convex Polyhedra. Springer, Berlin (2005) · Zbl 1133.52301
[4] Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998) · Zbl 0912.14023
[5] Minkowski, H.: Allgemeine Lehrsätze über die convexen Polyeder. Gött. Nachr. 198-219 (1897) · JFM 28.0427.01
[6] Minkowski, H.: Gesammelte Abhandlungen von Hermann Minkowski. Band I. Teubner, Leipzig (1911) · JFM 42.0023.03
[7] Panina, G, A.D. alexandrov’s uniqueness theorem for convex polytopes and its refinements, Beitr. Algebra Geom., 49, 59-70, (2008) · Zbl 1145.52007
[8] Sullivan, D.: Combinatorial invariants of analytic spaces. In: Proceedings of Liverpool Singularities—Symposium, I. Department of Pure Mathematics, University of Liverpool 1969-1970, 165-168 (1971)
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