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 -