Any round convex body between a simplex and a ball?

Question

Fritz John's ellipsoid theorem gives the minimal ball containing a given convex body $K$. Moreover, we have for $m$ points in boundary
$$\tag{$A$}\label{A}\sum_m c_iu_iu_i^{T} = I_n,$$
where $I_n$ is the identity matrix, $c_i$ some constants, and $u_i$ vectors representing points in $\partial(K) \cap \partial(B_n(0,1))$. Now, I would like to see conditions that quarantee that $u_i$ where $i=1,\dotsc m$ contain $n$ orthonormal vectors. I try to solve it like this. Let $S_n$ be the regular simplex. I can assume also that the ellipsoid is the unit ball. Clearly I can find one common point on the unit sphere just by rotating the simplex. First I let the $u_1 \in \partial(B(0,1)) \cap \partial(K)$ and $v_1 \in \partial(B(0,1)) \cap \partial(S_n)$ be such that $u_1 = v_1.$ Then I use the minimization procedure for the minimal ellipsoid containing $K \cap S_n \cap B(0,1)$. So the point $u_1$ will be preserved. This means that the minimal ball containing $K \cap S_n \cap B(0,1)$ will be unit ball. (EDIT: A priori the minimal ellipsoid will not necessarily be a ball, and this seems to work only if the resulting body is a ball.) In fact even this new convex body $K \cap S_n \cap B(0,1)$ will contain at least $n$ points on the boundary of $B(0,1)$.As an application I think of the following. The entries of the covariance matrix $Cov K$ of a convex body $K$ are defined as \begin{equation} \label{last} (a_{ij}) = \frac{\int_K x_ix_j}{\lvert K\rvert} - \frac{\int_K x_i}{|K|}\frac{\int_K x_j}{\lvert K\rvert}. \end{equation} We define the isotropic constant of any convex body $K$ in scaling invariant way using \begin{equation} \label{last2} L^{2n}_{K} :=\frac{\operatorname{Det}(Cov{K})}{|K|^2}. \end{equation}
Now, if I assume that the minimal ellipsoid containing the convex body $K$ is in the unit ball I have $$L^2_K = \int_K \frac{x_ix_i}{|K|^{1+2/n}} \leq |K|^{-2/n}.$$ But if the simplex $S_n \subset K,$ then I have $$L^2_K \leq |K|^{-2/n} \leq |S|^{-2/n} = (n!)^{2/n}.$$ Perhaps the simplex is the extremal case.