Talk:Relative interior

Possible error

The equivalent definition seems incorrectly stated. The relative interior as given would appear to be the entire set, since any point in the set can be written as a combination of half itself and half itself again, meeting the criteria. Also, it cannot be straightforwardly adapted to be correct, because requiring the other two points be different from the original does not explain how the relative interior of a single point is the point itself.— Preceding unsigned comment added by 71.243.16.79 (talk) 09:10, 23 February 2012 (UTC)[reply]

While I will look for a reference for that secondary definition, your argument against it isn't correct. This definition says:

x\in \operatorname {relint} C

(convex) if and only if for every

y\in C

there exists a

z\in C

such that

x

lies on the line between

y

and

z

and

x\not \in \{y,z\}

. So there is the error that for some point

y\in C

then

x=y

and that would automatically mean

x\not \in \operatorname {relint} C

, but otherwise you need more than just the point to lie on a single line but instead on all possible lines in the set. I hope I explained that well. Zfeinst (talk) 16:20, 23 February 2012 (UTC)[reply]

You were right that this formula was incorrect. I have found 2 books with a correct definition for finite dimensional convex sets. Zfeinst (talk) 18:09, 23 February 2012 (UTC)[reply]

Relative interior is much more general concept and it can defined for any topological space. It has nothing to do with vector/affine spaces in particular. This needs to be explained. -- David — Preceding unsigned comment added by David Pal (talk • contribs) 05:24, 4 February 2011 (UTC)[reply]

I don't understand why any metric will work for the epsilon balls. If we take for example the discrete metric, we will find that B_{\epsilon}(x) = {x}, thus we can't speak of an interior at all. If we take for example the n-sphere with edge, than the relative interior will be simple the sphere without it's edge, but with the discrete metric it will be the whole sphere. — Preceding unsigned comment added by 129.125.21.97 (talk) 13:20, 22 June 2011 (UTC)[reply]