1. Given one vertex point in $\{0,1\}^n$ and generators of symmetry group of a regular convex polytope whose vertices are all in $\{0,1\}^n$ is the embedding of the polytope in $\Bbb R^n$ unique?

2. If so can we find the John's ellipsoid of this polytope in polynomial time?

1. Do the symmetry group generators of a regular convex polytope and a marked $\{0,1\}^n$ vertex point suffice to embed the polytope uniquely with $\{0,1\}^n$ vertex set?

2. If so can we find the John's ellipsoid of this polytope in polynomial time?

Given one vertex point in $\{0,1\}^n$ and generators of symmetry group of a regular convex polytope whose vertices are all in $\{0,1\}^n$ is the embedding of the polytope in $\Bbb R^n$ uniquely definied?

1. Given one vertex point in $\{0,1\}^n$ and generators of symmetry group of a regular convex polytope whose vertices are all in $\{0,1\}^n$ is the embedding of the polytope in $\Bbb R^n$ unique?

2. If so can we find the John's ellipsoid of this polytope in polynomial time?

Given one vertex point in $\{0,1\}^n$ and generators of symmetry group of a regular convex polytope whose vertices are all in $\{0,1\}^n$ is the polytope uniquely definied?

Given one vertex point in $\{0,1\}^n$ and generators of symmetry group of a regular convex polytope whose vertices are all in $\{0,1\}^n$ is the embedding of the polytope in $\Bbb R^n$ uniquely definied?