This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes.

For a given $\delta$, let $n_\delta$ be the number of faces needed to approximate any convex body by a **contained** convex polytope with at most $n_\delta$ faces and Hausdorff distance at most $\delta$.
I'm interested in **non-asymptotic** **upper-bounds** on this number $n_\delta$ of faces.

In the following survey: http://link.springer.com/article/10.1007/s10958-008-9144-x, the author gives in (6) a non asymptotic upper bound, on the number of vertices $n_{\delta,v}$ needed to approximate a convex body by a polytope with Hausdorff distance at most $\delta$: $$\delta \leq \frac{C}{n_{\delta,v}^{\frac{2}{d-1}}} $$ He then states "Certainly the same estimates hold for" approximation by polytopes contained in/containing the convex body, and also replacing vertices by faces.

So when taking "faces" and "contained in", this is exactly the result I want. Unfortunately, there is no clear reference given to such results.

Since this problem is really far from my field, I don't have access to many sources through my department's subscriptions to scientific journals and through my libray. If I knew a precise reference, it would be no problem for me to ask my department to purchase the precise reference or my library to borrow the precise book, but it is very difficult for me to find such a precise reference.

So I'd be very grateful if anyone could point a precise reference to this result out to me.

J. Math. Sci.(N.Y.) 153 (2008), no.6, 727--762. $\endgroup$