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I have written a significant expansion of this page, but after several hours mozilla died on me, and have no strength to start it over again right now, so I just leave a split-and-paste version for now. The text has a number of inaccuracies, to be fixed. Twri ( talk) 18:13, 3 October 2008 (UTC)
Another curiosity, I guess the definition by bounding planes still works for lower dimensional polytopes, like a 2d face in 3-space? Like Image:Permutohedron order 3.svg is constrained by 8 hyperplanes. The constraining hyperplane inequalities can be doubly defined to reduce two opposite half-spaces as the common plane. Similarly an edge in 3-space can be defined as 4 bounding hyperplanes, two for a common line, and two for the endpoints. Well, no idea what this is worth. Of course it's always better to define a lower dimensional element by a parametric subspace anyway. Tom Ruen ( talk) 01:04, 4 October 2008 (UTC)
Does anyone have good references for treatment of convex polytopes as an objects in the projective space? This treatment would remove the distinction of bounded vs. unbounded polytopes: since an infinite ridge connects to a point in the projective space, so any convex polytope may be defined in terms of convex combinations of points in the projective space (instead of the Finite Basis Theorem), if I am not mistaken. Twri ( talk) 16:55, 6 October 2008 (UTC)
The link on ext points to the exterior of a set, where in this context it means the extreme points. The extreme points of a set X is the smallest set S such that the convex hull of S equals X. —Preceding unsigned comment added by 72.85.2.217 ( talk) 23:44, 22 October 2009 (UTC)
I was thinking a bounded convex polytope in R3 would be an ordinary sphere (2-sphere) topologically. The number 3 is odd, but the Euler characteristic of a sphere (2-sphere) is 2 rather than 0, right? Or maybe I am misunderstanding something. -- Keith111 ( talk) 17:06, 9 November 2010 (UTC)
The boundary of a convex polytope in Rn is an (n − 1)-dimensional manifold that is topologically a sphere, that is, an (n − 1)-sphere, and has the Euler characteristic given. The polytope itself is an n-ball, and has Euler characteristic 1. Does this clarify the issue? I'll go fix the article to say something like that. — David Eppstein ( talk) 17:02, 23 February 2011 (UTC)
In the section "Face lattice", the terminology "face" and "facet" are not actually defined, and I think it would add to the article if the definitions were added. Lavaka ( talk) 20:53, 18 November 2010 (UTC)
The definition of face (i.e., A face of a convex polytope is any intersection of the polytope with a halfspace such that none of the interior points of the polytope lie on the boundary of the halfspace) is not quite right. A face of the polytope is either the polytope itself, or the intersection of the polytope with a set of its supporting hyperplanes (previously defined in the article.) — Preceding unsigned comment added by Danpmoore ( talk • contribs) 16:44, 18 August 2017 (UTC)
The homeomorphism with a topological ball does not seem to be discussed in any easily accessible literature on polytopes that I know of. Can anybody provide any such reference? (This is for my own interest, I am assuming the idea is well enough sourced in the more advanced literature). — Cheers, Steelpillow ( Talk) 12:41, 30 May 2011 (UTC)
The article currently states
"The finite basis theorem[2] is an extension of the notion of V-description to include infinite polytopes. The theorem states that a convex polyhedron is the convex sum of its vertices plus the conical sum of the direction vectors of its infinite edges."
What about the convex polyhedron
x+y+z <= 1 2x-y+3z <=2
This has no vertices but it is not a conic sum of direction vectors.
????
98.155.30.229 ( talk) 17:11, 25 July 2015 (UTC)
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This article is rated C-class on Wikipedia's
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I have written a significant expansion of this page, but after several hours mozilla died on me, and have no strength to start it over again right now, so I just leave a split-and-paste version for now. The text has a number of inaccuracies, to be fixed. Twri ( talk) 18:13, 3 October 2008 (UTC)
Another curiosity, I guess the definition by bounding planes still works for lower dimensional polytopes, like a 2d face in 3-space? Like Image:Permutohedron order 3.svg is constrained by 8 hyperplanes. The constraining hyperplane inequalities can be doubly defined to reduce two opposite half-spaces as the common plane. Similarly an edge in 3-space can be defined as 4 bounding hyperplanes, two for a common line, and two for the endpoints. Well, no idea what this is worth. Of course it's always better to define a lower dimensional element by a parametric subspace anyway. Tom Ruen ( talk) 01:04, 4 October 2008 (UTC)
Does anyone have good references for treatment of convex polytopes as an objects in the projective space? This treatment would remove the distinction of bounded vs. unbounded polytopes: since an infinite ridge connects to a point in the projective space, so any convex polytope may be defined in terms of convex combinations of points in the projective space (instead of the Finite Basis Theorem), if I am not mistaken. Twri ( talk) 16:55, 6 October 2008 (UTC)
The link on ext points to the exterior of a set, where in this context it means the extreme points. The extreme points of a set X is the smallest set S such that the convex hull of S equals X. —Preceding unsigned comment added by 72.85.2.217 ( talk) 23:44, 22 October 2009 (UTC)
I was thinking a bounded convex polytope in R3 would be an ordinary sphere (2-sphere) topologically. The number 3 is odd, but the Euler characteristic of a sphere (2-sphere) is 2 rather than 0, right? Or maybe I am misunderstanding something. -- Keith111 ( talk) 17:06, 9 November 2010 (UTC)
The boundary of a convex polytope in Rn is an (n − 1)-dimensional manifold that is topologically a sphere, that is, an (n − 1)-sphere, and has the Euler characteristic given. The polytope itself is an n-ball, and has Euler characteristic 1. Does this clarify the issue? I'll go fix the article to say something like that. — David Eppstein ( talk) 17:02, 23 February 2011 (UTC)
In the section "Face lattice", the terminology "face" and "facet" are not actually defined, and I think it would add to the article if the definitions were added. Lavaka ( talk) 20:53, 18 November 2010 (UTC)
The definition of face (i.e., A face of a convex polytope is any intersection of the polytope with a halfspace such that none of the interior points of the polytope lie on the boundary of the halfspace) is not quite right. A face of the polytope is either the polytope itself, or the intersection of the polytope with a set of its supporting hyperplanes (previously defined in the article.) — Preceding unsigned comment added by Danpmoore ( talk • contribs) 16:44, 18 August 2017 (UTC)
The homeomorphism with a topological ball does not seem to be discussed in any easily accessible literature on polytopes that I know of. Can anybody provide any such reference? (This is for my own interest, I am assuming the idea is well enough sourced in the more advanced literature). — Cheers, Steelpillow ( Talk) 12:41, 30 May 2011 (UTC)
The article currently states
"The finite basis theorem[2] is an extension of the notion of V-description to include infinite polytopes. The theorem states that a convex polyhedron is the convex sum of its vertices plus the conical sum of the direction vectors of its infinite edges."
What about the convex polyhedron
x+y+z <= 1 2x-y+3z <=2
This has no vertices but it is not a conic sum of direction vectors.
????
98.155.30.229 ( talk) 17:11, 25 July 2015 (UTC)
Hello fellow Wikipedians,
I have just modified one external link on Convex polytope. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
This message was posted before February 2018.
After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than
regular verification using the archive tool instructions below. Editors
have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
RfC before doing mass systematic removals. This message is updated dynamically through the template {{
source check}}
(last update: 5 June 2024).
Cheers.— InternetArchiveBot ( Report bug) 18:48, 12 August 2017 (UTC)