convex geometry – the John ellipsoid of a polytope

Assume that $ X $ is $ mathbb R ^ n $ with a certain polyhedral standard, that is to say the unit ball of $ X $ is a $ n $polytope to size. Suppose the ellipsoid John of $ X $ is a Euclidean ball that touches all sides of the ball $ X $.

Is it true that we can find $ n $ orthonormal vectors that make the ball of $ X $ invariant under swap their order and change sign? (That is, the symmetry group of $[-1,1]^ n $ is a subgroup of the symmetry group of the ball of $ X $?)

Convex geometry – Volume of caps of a polytope

Let K $ to be a polytope $ mathbb R ^ d $, explode it from a postman $ lambda> $ 0. For a unit vector $ u in mathbb S ^ {d-1} $, $ lambda K $ has 2 support hyperplanes $ H_1 $ and $ H_2 $ with corresponding outgoing normal vectors $ u $ and $ -u $. I only take into account $ u $is such that $ H_1 $ and $ H_2 $ have "good" position, which says that they are not parallel to any face of K $, or equivalently, they contain only one vertex of $ lambda K $.

Consider another parallel hyperplane $ H_t $ such as the distance between her and $ H_1 $ is $ t $ (of course $ t $ is smaller than the width of $ lambda K $ in direction $ u $). So $ H_1 $ and $ H_t $ set a ceiling of $ lambda K $.

My question is:
Is there a chance that we can estimate the volume of this ceiling, in terms of $ u, lambda $ and $ t $?

The easiest case is when $ t $ is less than the distance from the nearest vertex of $ lambda K $ at $ H_1 $, then the volume is $ frac {1} {d} to ^ d $, or $ a $ is a constant depends on $ u $.

I mean, in general, it's very difficult to calculate the volume of similar definition plugs K $. But if we blow up K $, some factors can be ignored when $ lambda $ is big enough, so I hope that there is a limit on the size of the cork volume.