## 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.