Talk:Polytope
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Does anyone know who the 'I' is that remarked at the bottom of this article? The history is lost with the change of software. ---
Rather than restrict ourselves to ASCII art, could someone please draw these figures in a graphics program and upload them? I would, but I know nothing about the subject and can't make heads or tails of the existing depictions. - user:Montrealais
Obviously needs an edit.
Charles Matthews 09:33 25 Jun 2003 (UTC)
On a closer inspection: is polytope just being used here for simplicial complex embedded in Euclidean space? Is there some condition too that makes it a manifold (or not)?
Charles Matthews 12:36 25 Jun 2003 (UTC)
Mathematical nonsense removed.
- Roughly speaking this is the set of all possible weighted averages, with weights going from zero to one, of the points. These points turn out to be the vertices of their convex hull. When the points are in general position (are affinely independent, i.e., no s-plane contains more than s + 1 of them), this defines an r-simplex (where r is the number of points).
Mikkalai 08:30, 1 Mar 2004 (UTC)
The "mathematical nonsense" should be rewritten and put in an article on convex hulls, if it hasn't already
mike40033 11:20, 1 Mar 2004 (GMT+0800)
[edit] half spaces & convex hulls
I think there's an error here:
One special kind of polytope is a convex polytope, which is the convex hull of a finite set of points. Convex polytopes can also be represented as the intersection of half-spaces.
How can this be simultaneously true? Consider a single half-space: note that it is certainly convex. Of what finite set of points is this polytope the convex hull?
My understanding, from the reference given below, is that
A (convex) polyhedron in Rk is defined to be the intersection of some finite number of half spaces in Rk. Bounded polyhedra are called polytopes. (A polytope can be definted equivalently as the convex hull of a finite point set in Rk).
Note that this agrees with Wikipedia's article on polyhedron.
If there are no arguments, I will edit to reflect this definition.
Reference: Dobkin, D. & Kirkpatrick, D., "A Linear Algorithm for Determining the Separation of Convex Polyhedra," Journal of Algorithms 6, 381-392 (1985).
-Alem
- I don't know if the term 'polytope' consistently refers to bounded polytopes (which is the definition you have here). Some papers refer to "unbounded polytopes" where the bounding halfspaces enclose an unbounded region. I don't think the term "polyhedron" generally refers to arbitrary dimensions; it usually refers only to R3. But you're right, there's a hole in the current definition which needs to be addressed.—Tetracube 21:18, 13 September 2006 (UTC)
