I'm trying to use the Qhull program to solve for convex hull. It seems to only take points as inputs. But my convex hull is so far only defined by the equation, i.e. ${\displaystyle \mathrm {Conv} (S)=\left\{\left.\sum _{i=1}^{|S|}\alpha _{i}x_{i}\ \right|(\forall i:\alpha _{i}\geq 0)\wedge \sum _{i=1}^{|S|}\alpha _{i}=1\right\}.}$
All the $\bf{x_{i}}$ vectors are known.
So can I just use $x_{i}$ as the points? are they the same as the vertices? or should I be randomly sample a bunch of points that fits in this equation, and then feed those points into the qhull program?
The equation you gave is just the characterization of the convex hull of a set of points as the set of convex combinations of the points. So you can just give the points to Qhull.