Talk:Convex polytope

New article

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)

Sorry to hear that. It must have been frustrating. Something similar happened to me before, too: I was in the middle of a massive rewrite of an article, when I hit the wrong key and closed the browser tab. Several hours of work vanished into the bit bucket. Nowadays I try not to make too big an edit before saving. Anyway, thanks for splitting up this article from polytope; I think polytope works better this way.—Tetracube (talk) 20:12, 3 October 2008 (UTC)

Hmmmm... I can't really add anything here, but curious on a definition, or specifically whether a polytope is considered a topological object, or as representing a geometric interior. This article talks only about the geometric interior properties.

... Okay, I added a short section for topology! Tom Ruen (talk) 22:28, 3 October 2008 (UTC)

Bounding planes

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)

This is equivalent, of course, to setting some rows of the linear system to equalities instead of inequalities. Komei Fukuda's cddlib allows you to specify some rows of the matrix to be equalities rather than inequalities, thereby handling the case when the polytope is confined to an affine subspace. You could also think of such polytopes as being facets of a larger polytope which is full-dimensioned; after all, a polytope's face is defined by the set of points satisfying equality for one or more of rows in the linear system.—Tetracube (talk) 03:16, 4 October 2008 (UTC)

Projective space

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)

Vertex representation

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)

Error in the article?

"All bounded convex polytopes in Rⁿ, being topological (n − 1)-spheres, have an Euler characteristic of 0 for odd n and 2 for even n."

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)

Yes, this seems to be backwards. —David Eppstein (talk) 18:26, 9 November 2010 (UTC)

Face lattice

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)