A centrally symmetric convex polyhedron in $\Bbb R^n$ shifted from the origin with possibly $e^{\alpha n}$ number of vertices at some $\alpha>0$ has an unique ellipsoid of maximum volume called John's ellipsoid inscribed in it with barycenter of the polyhedron agreeing with center of the ellipsoid.

Obtain an uniform sample of $O(n^c)$ vertices at some fixed $c\geq2$.

1. Can we apply central limit theorem to approximate barycenter by mean of sample vertices?

2. Since ellipsoids have only $O(n^2)$ parameters does this sample give approximation to the ellipsoid?

A centrally symmetric convex polyhedron in $\Bbb R^n$ shifted from the origin with possibly $e^{\alpha n}$ number of vertices at some $\alpha>0$ has an unique ellipsoid of maximum volume called John's ellipsoid inscribed in it with barycenter of the polyhedron agreeing with center of the ellipsoid.

Obtain an uniform sample of $O(n^c)$ vertices at some fixed $c\geq2$.

1. Can we apply central limit theorem to approximate barycenter by mean of sample vertices?

2. Since ellipsoids have only $O(n^2)$ parameters does this sample give a constant fraction approximation of each axis to the ellipsoid?

A centrally symmetric convex polyhedron in $\Bbb R^n$ shifted from the origin with possibly $e^{\alpha n}$ number of vertices at some $\alpha>0$ has an unique ellipsoid of maximum volume called John's ellipsoid inscribed in it with barycenter of the polyhedron agreeing with center of the ellipsoid.

1. Does getting an uniform sample of $O(n^c)$ vertices for some fixed $c\geq2$ suffice to approximate the ellipsoid up to $\epsilon$ factor in each axis provided we know that the barycenter agrees with center of the ellipsoid and we know the center within some neighborhood?

I am asking this since the ellipsoid has only $O(n^2)$ parameters.

A centrally symmetric convex polyhedron in $\Bbb R^n$ shifted from the origin with possibly $e^{\alpha n}$ number of vertices at some $\alpha>0$ has an unique ellipsoid of maximum volume called John's ellipsoid inscribed in it with barycenter of the polyhedron agreeing with center of the ellipsoid.

Obtain an uniform sample of $O(n^c)$ vertices at some fixed $c\geq2$.

1. Can we apply central limit theorem to approximate barycenter by mean of sample vertices?

2. Since ellipsoids have only $O(n^2)$ parameters does this sample give approximation to the ellipsoid?

A centrally symmetric convex polyhedron in $\Bbb R^n$ shifted from the origin with possibly $e^{\alpha n}$ number of vertices at some $\alpha>0$ has an unique ellipsoid of maximum volume called John's ellipsoid inscribed in it with barycenter of the polyhedron agreeing with center of the ellipsoid.

1. Does getting an uniform sample of $O(n^c)$ vertices for some fixed $c\geq2$ suffice to approximate the ellipsoid up to $\epsilon$ factor in each axis provided we know that the barycenter agrees with center of the ellipsoid and we know the center within some neighborhood?

I am asking this since the ellipsoid has only $O(n^2)$ parameters.

A centrally symmetric convex polyhedron in $\Bbb R^n$ shifted from the origin with possibly $e^{\alpha n}$ number of vertices at some $\alpha>0$ has an unique ellipsoid of maximum volume called John's ellipsoid inscribed in it with barycenter of the polyhedron agreeing with center of the ellipsoid.

1. Does getting an uniform sample of $O(n^c)$ vertices for some fixed $c\geq2$ suffice to approximate the ellipsoid up to $\epsilon$ factor in each axis provided we know that the barycenter agrees with center of the ellipsoid and we know the center within some neighborhood?

I am asking this since the ellipsoid has only $O(n^2)$ parameters.