TY - JOUR

T1 - Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?

AU - Alexandrov, Victor

PY - 2018/4/1

Y1 - 2018/4/1

N2 - 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.

AB - 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.

KW - Analytic set

KW - Convex polyhedron

KW - Euclidean space

KW - Perimeter of a face

UR - http://www.scopus.com/inward/record.url?scp=85037700657&partnerID=8YFLogxK

U2 - 10.1007/s12188-017-0189-y

DO - 10.1007/s12188-017-0189-y

M3 - Article

AN - SCOPUS:85037700657

VL - 88

SP - 247

EP - 254

JO - Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg

JF - Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg

SN - 0025-5858

IS - 1

ER -