TY - JOUR

T1 - Concurrent normals problem for convex polytopes and Euclidean distance degree

AU - Nasonov, I.

AU - Panina, G.

AU - Siersma, D.

N1 - Export Date: 18 November 2024 Текст о финансировании 1: This work was done under support of the grant No. 075-15-2022-289 for creation and development of Euler International Mathematical Institute.

PY - 2024/11/6

Y1 - 2024/11/6

N2 - It is conjectured since long that for any convex body P⊂Rn there exists a point in its interior which belongs to at least 2n normals from different points on the boundary of P. The conjecture is known to be true for n=2,3,4. We treat the same problem for convex polytopes in R3. It turns out that the PL concurrent normals problem differs a lot from the smooth one. One almost immediately proves that a convex polytope in R3 has 8 normals to its boundary emanating from some point in its interior. Moreover, we conjecture that each simple polytope in R3 has a point in its interior with 10 normals to the boundary. We confirm the conjecture for all tetrahedra and triangular prisms and give a sufficient condition for a simple polytope to have a point with 10 normals. Other related topics (average number of normals, minimal number of normals from an interior point, other dimensions) are discussed. © The Author(s), under exclusive licence to Akadémiai Kiadó, Budapest, Hungary 2024.

AB - It is conjectured since long that for any convex body P⊂Rn there exists a point in its interior which belongs to at least 2n normals from different points on the boundary of P. The conjecture is known to be true for n=2,3,4. We treat the same problem for convex polytopes in R3. It turns out that the PL concurrent normals problem differs a lot from the smooth one. One almost immediately proves that a convex polytope in R3 has 8 normals to its boundary emanating from some point in its interior. Moreover, we conjecture that each simple polytope in R3 has a point in its interior with 10 normals to the boundary. We confirm the conjecture for all tetrahedra and triangular prisms and give a sufficient condition for a simple polytope to have a point with 10 normals. Other related topics (average number of normals, minimal number of normals from an interior point, other dimensions) are discussed. © The Author(s), under exclusive licence to Akadémiai Kiadó, Budapest, Hungary 2024.

KW - 52B70

KW - bifurcation

KW - Morse theory

KW - polyhedra

UR - https://www.mendeley.com/catalogue/996c529f-956f-3d82-9709-7180adb9f5df/

U2 - 10.1007/s10474-024-01483-2

DO - 10.1007/s10474-024-01483-2

M3 - статья

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

SN - 0236-5294

ER -