# simplicial polytopes and convex hull

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?

• The key is "points that are in general position". Note that the four points required to generate the square are four coplanar points - not general position... en.wikipedia.org/wiki/General_position Jul 22, 2019 at 22:05
• @HughDenoncourt: thanks, I didn't know the expression had a special meaning. I interpreted it as "any". If you convert the comment to an answer, I'll accept it. Jul 22, 2019 at 22:08
• Cool, thanks. Good luck; polytopes are fun. Jul 23, 2019 at 0:13

## 2 Answers

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.

Read "general posistion" as "not satisfying more affine dependencies than absolutely necessary". Or in other words:

• no two vertices on the same point
• no three vertices on the same line
• no four vertices on the same plane
• ...

The third points is not valid for the square based pyramid, as there are four points on a common plane.

And in general, if you have a $$d$$-dimensional face that is not a $$d$$-simplex, then this face contains at least $$d+2$$ vertices, and these all lie in a common $$d$$-dimensional affine space, in contradiction to "general position".