On page 120 of his *Basic Topology*, Armstrong defines the $k$-simplex in $\mathbb{E}^n$ with verices $v_0,\ldots,v_k$ to be complex hull of said vertices. (A similar definition is given on Wikipedia).

Here's what bothers me: Given a $k$-simplex $\sigma$ ($k>0$) so defined, and $m$ other points (all distinct from each other and from the vertices of $\sigma$) in $\sigma$, the above definition says that the convex hull $\sigma'$ of the $k+1$ vertices plus the $m$ additional points is a $k+m$-simplex. But $\sigma=\sigma'$. So, $\sigma$ is both a $k$- and $k+m$-simplex.

Am I missing something?

general position. Are you sure there is no such requirement in Armstrong's text? I guess if not, then one can speak of a "degenerate k-simplex", in the same sense that a line (1-simplex) is a degenerate triangle (2-simplex). – Faisal Apr 8 '11 at 17:42convex hullnot thecomplex hull. I've never seen the term complex hull before. – Ryan Budney Jun 11 '11 at 23:54