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Polytope with most faces

MathOverflow Asked by zjs on December 1, 2021

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.)

One Answer

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 $Delta(m,n)$.

Answered by Joshua Mundinger on December 1, 2021

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