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.)



Polytope with most faces
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 $\Delta(m,n)$.


