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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?

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    $\begingroup$ 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 $\endgroup$ Jul 22, 2019 at 22:05
  • $\begingroup$ @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. $\endgroup$
    – Matteo
    Jul 22, 2019 at 22:08
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    $\begingroup$ Cool, thanks. Good luck; polytopes are fun. $\endgroup$ Jul 23, 2019 at 0:13

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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.

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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".

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