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The volume ratio of a convex body $K\subset \mathbb{R}^{n}$ is $v_r(K) = \inf_{\mathcal{E}\subset K} \left(\frac{Vol(K)}{Vol(\mathcal{E})}\right)^{1/n}$ where the infimum run over ellipsoids included in K

By using John's Theorem, Keith Ball showed that the cube has largest volume ratio among symmetric convex bodies (of order $\sqrt{n}$).

Define dually $v_r^\circ(K) = \inf_{K\subset \mathcal{E}} \left(\frac{Vol(\mathcal{E})}{Vol(K)}\right)^{1/n}$ where the infimum run over ellipsoids containing K. By duality and using Bourgain-Milman theorem which says that the quantity $Vol(K)Vol(K^\circ)^{1/n}$ is upper and lower bounded on the class of convex bodies by two absolute constants, we see that $v_r^\circ(K)$ is up to constant maximised by the cube among symmetric convex bodies (I suspect it is indeed the maximum).

My question is : is there a quantitative stability result in terms of the number of faces ? That is, if $K$ has $k$ faces is there a result looking like

$$v_r(K) \geq c_0 \frac{\sqrt{n}}{1+\log(k/n)} $$

Or dually

$$v_r^\circ(K) \leq c_0 (1+\log(k/n)) $$ something like that ?

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The cube has a bounded outer volume ratio (which you denote $v_r^\circ$) while the cross-polytope (=$\ell_1$ ball, or octahedron) has an outer volume ratio of order $\sqrt{n}$. Actually the cross-polytope maximizes the outer volume ratio among symmetric convex bodies, this is the content of the Remark at the end of §1 in Ball, K. M., Volume ratios and a reverse isoperimetric inequality, J. Lond. Math. Soc., II. Ser. 44, No. 2, 351-359 (1991). ZBL0694.46010. ; I don't know if a more detailed argument has be written somewhere.

Convex bodies with few vertices have a large outer volume ratio: if a $n$-dimensional polytope has $k$ vertices, then $$ v_r^\circ(K) \geq c \sqrt{\frac{n}{1+\log(k/n)}}; $$ or dually, if a $K$ has $k$ facets, $$ v_r(K) \geq c \sqrt{\frac{n}{1+\log(k/n)}}. $$ This estimates are well-known; they follow for example from Proposition 6.3 and Remark 6.4 in Aubrun, Guillaume; Szarek, Stanisław J., Alice and Bob meet Banach. The interface of asymptotic geometric analysis and quantum information theory, Mathematical Surveys and Monographs 223. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3468-7/hbk; 978-1-4704-4172-2/ebook). xxi, 414 p. (2017). ZBL1402.46001.

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