Timeline for Number of vertices in a polyhedron
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Apr 15, 2021 at 9:29 | comment | added | Turbo | This is impossible if the convex combination is a linear combination of two or more vertices as $x_{1,1},\dots,x_{2,m_2},z$ is a vertex point. Hence there is a $z_1$ satisfying the property $(x_1,z_1)=(x_{1,1},\dots,x_{1,m_1},z_1)$ is a vertex of the first polyhedron. How about $(x_2,z_2)$ and $(x_1,z_1)$ are not vertices in their respective polyhedron? I think $v_1v_2$ holds and I do think above strategy works but not entirely positive. | |
| Apr 15, 2021 at 9:29 | comment | added | Turbo | Pick one (say $(x_1,z_1')$) without loss of generality and again by Caratheodory's theorem $(x_1,z_1')$ is a convex combination of $m_1+1$ vertex points on the face containing $(x_1,z_1')$ after projecting the vertices to $z_1'$ in the last coordinate. Now project out the last coordinate and lift to dimension $m_1+m_2+1$ by adding $(x_2,z)$ in the end. This implies $x_{1,1},\dots,x_{2,m_2},z$ is the convex combination of the same $m_1+1$ projected (to $z_1'$) vertex points after lifting to $m_1+m_2+1$ dimensions by adding $(x_2,z)$ in the end. | |
| Apr 15, 2021 at 9:28 | comment | added | Turbo | On vertices following is my thought suppose $x_{1,1},\dots,x_{2,m_2},z$ is a vertex of third polyhedron and $(x_2,z_2)=(x_{2,1},\dots,x_{2,m_2},z_2)$ is a vertex of second polyhedron at some $z_2$ and $(x_1,z_1)=(x_{1,1},\dots,x_{1,m_1},z_1)$ is not a vertex of first polyhedron at any z_1. Now $(x_1,z_1)$ is a convex combination of $m_1+2$ vertex points in first polyhedron by Caratheodory's theorem and by convexity there are two $z_1'$ and $z_1''$ at which $(x_1,z_1')$ and $(x_1,z_1'')$ is on the $m_1$ dimensional face of the first polyhedron. | |
| Apr 15, 2021 at 9:25 | comment | added | Stefan Kohl♦ | Which purpose did it serve to delete and undelete this question several times in a row? -- It certainly does not help you to get good answers. | |
| Apr 6, 2021 at 22:41 | history | undeleted | Turbo | ||
| Apr 6, 2021 at 13:50 | history | deleted | Turbo | via Vote | |
| Apr 6, 2021 at 13:49 | history | undeleted | Turbo | ||
| Apr 6, 2021 at 13:49 | history | deleted | Turbo | via Vote | |
| Apr 5, 2021 at 21:21 | history | edited | Turbo | CC BY-SA 4.0 |
added 17 characters in body
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| Apr 5, 2021 at 21:06 | history | undeleted | Turbo | ||
| Apr 5, 2021 at 18:42 | history | deleted | Turbo | via Vote | |
| Apr 5, 2021 at 17:31 | history | undeleted | Turbo | ||
| Apr 5, 2021 at 17:30 | history | deleted | Turbo | via Vote | |
| Apr 5, 2021 at 17:25 | history | edited | Turbo | CC BY-SA 4.0 |
[Edit removed during grace period]
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| Apr 5, 2021 at 17:15 | history | asked | Turbo | CC BY-SA 4.0 |