Let $P$ be the polytope obtained as the convex hull of $n$ points in $\mathbb R^d$. This is the $V$-representation of $P$. Note that $P$ can also be represented as an intersection of closed half-spaces in $\mathbb R^d$. This is the $H$-representation of $P$.



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> **Question.** *In terms of $n$ and $d$, what is a good upper-bound on the number of half-spaces in the smallest $H$-representation of $P$ ?*
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Optimal number of half-spaces in the $H$-representation of the convex hull of $n$ points in $\mathbb R^d$
By the [Upper Bound Theorem](https://en.wikipedia.org/wiki/Upper_bound_theorem), the maximum number of $(d-1)$-dimensional faces of an $n$-vertex polytope is achieved by the cyclic polytopes. This number can be explicitly written via the [Dehn-Sommerville equations](https://en.wikipedia.org/wiki/Dehn%E2%80%93Sommerville_equations).


