What are the least number of extremal points on a polytope?

Question

Given a polytope $P$ embedded in $\mathbb{R}^D$. I can't prove the property that any facet of the polytope will have at least $D$ extremal points lying on it. I can see it for the case $D=2$ but not in general.

Answer 1

You need an additional assumption; line segments count as polytopes, but they always have facets containing exactly $1$ vertex, regardless of $D$.

So, assuming you mean a polytope $P \subseteq \Bbb R^D$ that is $D$-dimensional, the basic idea is that you need at least $k$ points to determine an affine subspace of dimension $k-1$. Stop reading for a moment and think about how the claim follows from this, and if necessary, why the claim is true.

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For more detail: When we say a polytope is dimension $D$, that means the affine span of its vertices is $D$-dimensional. Any facets of a $D$-dimensional polytope must be $(D-1)$-dimensional polytopes, and thus have a $(D-1)$-dimensional affine span. That means each facet needs at least $D$ vertices.