I've seen some authors define the 2-simplex as the boundary of a triangle and others define it including the interior of the triangle (i.e. filling in the triangle). Does this distinction matter? Are they two different definitions?
Same question goes for the 3-simplex regarding the interior of the tetrahedron.
Thanks.
I suppose you're talking about a geometric 2-simplex. Now, I can't tell whether you've encountered the following definition, but this is the one I was introduced to and it helps me picture a 2-simplex very clearly so I think it might help.
Let $X \subset \mathbb{R}^{m}$ be a set of $n+1$ geometrically independent points. A geometrical n-simplex in $\mathbb{R}^{m}$ is the following set: $$ \Delta(X) := \{Q = \sum\limits_{P \in X} a_{P}P \ | \ \sum\limits_{P \in X} a_{P} = 1, a_{P} \geq 0 \ \forall P \in X \} $$
Basically, this is the set of affine linear combinations of points in $X$ with the additional property that each coefficient $a_{P}$ is greater than zero. If we choose $n=2$, this means that we are only considering the inside of the triangle whose vertices are the points in $X$ --and its border, of course --, instead of the whole affine 2-plane the vectors would span in $\mathbb{R}^{m}$.
If you allowed the coefficients to be negative, you could also obtain vectors in the affine 2-plane who point outside of the triangle.