zbMATH — the first resource for mathematics

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.

52B10 Three-dimensional polytopes
51M20 Polyhedra and polytopes; regular figures, division of spaces
Full Text: DOI
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.