There are many results how many vertices we need to approximate a convex body. It is known, for a given $\tau > 0$, we need $O(\tau^d)$ vertice of set $X = \{x_1, \dots, x_m\} \subset \mathbb{R}^d$ to make $K \subset conv(X) \subset (1+\tau) K $ holds for a convex body K where $conv(X)$ is the convex hull containing $X$ and $(1+\tau)K = \{(1+\tau) x \in \mathbb{R}^d : x \in K \}$. (Ref : Bronstein, Efim M. "Approximation of convex sets by polytopes." Journal of Mathematical Sciences 153.6 (2008): 727-762.)
However, I cannot find any result about polyhedron approximation. That is, I wonder how many closed half spaces we need to approximate a convex body K to make $K \subset \bigcap_{j=1}^m H_j^+ \subset (1+\tau) K$. One obvious bound comes from the upper bound theorem but I guess there are shaper bounds order of $O(\tau^d)$. Is there any reference about this issue?