Suppose I have a Lipschitz-continuous convex function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. I wish to approximate it on the unit ball by a piecewise-linear function $g:\mathbb{R}^n\rightarrow \mathbb{R}$ such that pointwise: $$\forall x : ||x||_2 \leq 1, |f(x) - g(x)| \leq \epsilon$$

Are there known bounds on the complexity of the piecewise linear function (i.e. the number of linear pieces needed) to achieve such an approximation, in terms of $\epsilon$ and $n$? I'm particularly interested in the dependence on the dimension $n$. Might the dependence be polynomial (or even linear?), or are there Lipschitz convex functions that require exponentially many linear pieces (in $n$) to approximate?