I'm tring to find the number of facets of the polyhedral cone $C=\text{cone}\{z_1,\cdots,z_{2^n}\}$, where $z_k$'s are the vertices of the hypercube $\|x-x_0\|_{\infty}\le r$. Specifically, I'm interested in the case where the hypercube is small enough and belongs to the nonnegative orthant.
I tried to compute the dual cone $$\begin{aligned} C^*&=\{y\mid z_k^Ty\ge 0,\forall 1\le k\le 2^n\}\\ &=\left\{y\mid \inf_{\|z-x_0\|_\infty\le r}z^Ty\ge 0\right\}\\ &=\{y\mid x_0^Ty-r\|y\|_1\ge 0\}. \end{aligned}$$ So $C^*$ is the projetion of the following polyhedron on the $y$-spcae. $$\begin{aligned} r1^Tu-x_0^Ty&\le 0\\ -u\le y&\le u \end{aligned}$$
If we can somewhat know the number of generators of $C^*$, we know the number of facets of $C$. An upper bound on the number of facets is also good to me.