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Fix $m,n \in \mathbb{N}$ with $m \ge n+1$. Take $m$ points in general position in $\mathbb{R}^n$ and let $P$ be their convex hull. What is the maximal number of (external, codimension-one) faces that $P$ can have, in terms of $m$ and $n$?

(Apologies if this is a well-known quantity.)

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    $\begingroup$ dual question: mathoverflow.net/questions/127423/… $\endgroup$
    – Jan Kyncl
    Commented Jul 22, 2020 at 23:56
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    $\begingroup$ See the answers at Jan's link. "Cyclic polytopes maximize the number of facets for a fixed number of vertices." For you, $m$ is the number of vertices. $\endgroup$
    – Joseph O'Rourke
    Commented Jul 23, 2020 at 0:32
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    $\begingroup$ Thank you both! This is extremely helpful, particularly the closed form. $\endgroup$
    – zjs
    Commented Jul 23, 2020 at 5:11

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The upper bound conjecture of Motzkin, made a theorem by McMullen in 1970, states that the highest number of facets among all polytopes with $m$ vertices in $\mathbb R^n$ is the number of facets of the cyclic polytope $\Delta(m,n)$.

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