Given two symmetric convex bodies $B, B'$ in ${\bf R}^d$, define the Banach-Mazur distance $d(B,B')$ between them to be the least constant $\tau \geq 1$ such that $$ B \subset TB' \subset \tau B$$ for some invertible linear transformation $T: {\bf R}^d \to {\bf R}^d$. (One should work with $\log \tau$ instead of $\tau$ if one wants this distance to actually be a (pseudo-)metric, but it seems more conventional to use $\tau$ in this subject.)
John's theorem for symmetric convex bodies is equivalent to the assertion that every symmetric convex body is within $\sqrt{d}$ of the Euclidean unit ball $B_{\ell^2}$. The bound of $\sqrt{d}$ is best possible, as can be seen by considering the unit cube. Thus, we get an extremely structured approximation to any symmetric convex body - namely, the Euclidean ball - at the cost of a somewhat weak bound ($\sqrt{d}$) on the accuracy of the approximation in Banach-Mazur distance.
I am interested in results that improve the accuracy of approximation, at the cost of weakening the structure of the approximant. An example of the type of result I have in mind is in
Barvinok, Alexander, Thrifty approximations of convex bodies by polytopes, Int. Math. Res. Not. 2014, No. 16, 4341-4356 (2014). ZBL1300.52007.
in which it is shown for instance that an arbitrary convex body can be approximated to error $\varepsilon \sqrt{d}$ in Banach-Mazur distance by a polytope with a polynomial number of vertices (roughly $d^{1/\varepsilon}$), or to error $1+\varepsilon$ with an exponential number of vertices (roughly $\varepsilon^{-d/2}$), as well as various interpolations between these two results.
My question is: are there other interesting classes of convex bodies (e.g., zonotopes) than just the unit ball (or ellipsoids) for which one has a John-type theorem with a significantly better approximation than $\sqrt{d}$? For instance, what if instead of approximating by a unit ball, one wishes to approximate by a convex body $B'$ that is only an epsilon away in measure from being a unit ball, in the sense that $$ \mathrm{vol}(B' \Delta B_{\ell^2}) \leq \varepsilon \mathrm{vol}(B_{\ell^2}):$$ can one significantly reduce the loss of $\sqrt{d}$ if one does this?
Alternatively, what if one is willing to pass to a reasonably high-dimensional slice of ${\bf R}^d$, e.g., an $\varepsilon d$-dimensional slice - does one get significant improvements to the constant $\sqrt{d}$ in John's theorem in this case? Of course by Dvoretsky's theorem one will eventually get approximation arbitrarily close to 1 if one takes a low enough dimensional slice (of dimension $\log d$ or so), but I would like to keep the dimension of the slice comparable to the dimension of the whole space.
