In Ziegler's Lectures on Polytopes (7th printing), on page 8, it is said that "the convex hull of any set of points that are in general position in $\mathbb{R}^d$ is a simplicial polytope", where "simplicial polytope" is defined slightly above as a "polytope, all of whose proper faces are simplices" (Ziegler uses "polytope" to mean "convex polytope").
I don't see how the statement about the convex hull can be correct. Take for example the pyramid with a square base in $\mathbb{R}^3$: one of its faces is definitively not a simplex.
Am I missing something?
Best Answer
The key is that "general position" is not the same as "arbitrary". The notion of general position depends upon context, but for this context, "general linear position" suffices.
A requirement for $n$ points to be in general linear position is that they be an affine basis for an $n - 1$ dimensional space. This is not satisfied by four points forming a square. The affine span of the four vertices is $2$-dimensional, not $3$, as required.