I am looking for a algorithm that returns the vertices of a polytope if provided with the set of hyperplanes that confine it.

In my special case the polytope is construced by the following constrains on $ \mathbb{R}^d$

- $ \left\| x \right\|$ = 1, $ x \in \mathbb{R}^d$
- $ 0<a_i \leq x_i\leq b_i \leq1$ , where $ x_i$ represents the i-th component of $ x$

This always generates a convex polytope which is confined by hyperplanes, if I am not mistaken. I would like to have an algorithm that returns a set of points P that are the vertices of the polytope if provided with the set of $ a_i$ ‘s and $ b_i$ ‘s

There is a special case where there exists no $ x$ that satifies this condition. Idealy this would be picked up on.

Thank you!