This is the isoperimetric type question. We know that in $\mathbb{R}^d$, balls are the sets that minimize the isoperimetric ratio $\frac{S^{d}}{V^{d-1}}$, where $S$ is the surface area and $V$ is the volumn.

Now consider the polytopes with $f$ faces. Lindelof's theorem says, among all proper convex polytopes in $\mathbb{R}^d$ with given exterior normals of the facets, it is precisely the polytopes circumscribed to a ball that have minimum isoperimetric quotient. This theorem can be found in http://link.springer.com/book/10.1007%2F978-3-540-71133-9, page 308, Theorem 18.4.

However, on Page 309, the author made a Corollary 18.2 that among all proper convex polytopes in $\mathbb{R}^d$ with a given number of facets, there are polytopes with minimum isoperimetric quotient and these polytopes are circumscribed to a ball.

Now my question is, I think the two claims above are different. To prove the Corollary 18.2, one has to prove the existence of the polytopes minimizing the isoperimetric constant among all polytopes circumscribed to a unit ball. I searched a lot of references, but I didn't find any proofs of such an existence. Is this an obvious result?