This is related to a research problem I'm working on relating to intersection graphs, though I'm not an expert in convex geometry (but learning!)
The hypothesis is this:
Let $n\geq 2$. Given $\varepsilon >0$ and $r > 0$, can we choose $N$ such that there exists a polytope $P_{n,N}$ in $\mathbb R^n$ with at most $N$ facets satisfying $$ B_n (r) \subset P_{n,N} \subset B_n (r+\varepsilon), $$ where $B_n(r)$ is the closed ball of radius $r$ in $\mathbb R^n$ centered at the origin.
I suspect the answer is yes if we let $N$ be large enough, but I'm not completely sure, and I'd also want to find out what the bounds on $N$ are as a function of $\varepsilon$ and $r$.
The literature on approximation of convex bodies by polytopes is vast, and it's hard for me to sift through all the results that could be related to this question, but the below seems like it should be a useful result in this direction (though I'm not sure how to get from it to my statement).
This paper says that there exists a constant $C$ such that for every $n\geq 2$ and $N\geq 10^n$, there exists a polytope $P_{n,N}$ in $\mathbb R^n$ with at most $N$ facets satisfying $$ \Delta(B_n(1),P_{n,N}) \leq C n^{-2/(n-1)}\operatorname{vol}_n(B_n(1)), $$ where $\Delta(A,B)$ is the volume (i.e., Lebesgue measure) of the symmetric difference of $A$ and $B$.
Does some related argument to this bound on volume get me where I want to go, and is there a simpler or more direct result existing about this question?
Edit: This paper on polytopal approximation of balls also seems useful, since it considers properties of polytopes inscribed and circumscribed by balls, whereas most papers I've read so far seem to only consider one case. However, I'm not sure it's what I want.
Edit 2: Since I messed up my quantifiers originally, an example might help alleviate any confusion. Let $n=2$. Say we have $\varepsilon = 0.1$ and $r=1$. My question amounts to asking: can we find a positive integer $N$ such that an $N$-sided convex polygon $P_{2,N}$ satisfies $B_2(1) \subset P_{2,N} \subset B_2(1.1)$? What is the smallest such $N$ such that this is satisfied?
Edit 3: Hm, maybe $N$ doesn't actually need to depend on $r$ since scaling a polygon doesn't change its number of facets.