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 $P$.

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 $P$ ?