# Polytope with most faces

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.)

• dual question: mathoverflow.net/questions/127423/… – Jan Kyncl Jul 22 at 23:56
• 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. – Joseph O'Rourke Jul 23 at 0:32
• Thank you both! This is extremely helpful, particularly the closed form. – zjs Jul 23 at 5:11

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 $$\Delta(m,n)$$.