Consider a set of $n$ points $x_i , i= 1 ... n$ belonging to some space $\mathbb{R}^m$. Given a point $p$ in the convex hull of the $x_i$ it is well known that we can represent $p$ as $ p = \sum _i ^ n \alpha _i x_i$ where $ \sum \alpha _i = 1$. It is also known that for $n=m+1$, the $\alpha _i$ are unique when $ \left\{ x_i -x_1 \right\} $ for $i=2...n$ are linearly independent. The question I have is: Is there some general algorithm to find the $\alpha _i$ such that all $\alpha _i \geq 0$. Also for the general case where $n=m+1$ need not be true, the $\alpha _i$ are not unique. Then is there some condition we can impose such that the $\alpha _i$ under this condition are unique? For example if we try and minimize the variance of the $\alpha_i$ then is the resulting set of $\alpha _i$ unique?
I found a similar question here: Write a point on/inside the convex hull of a set of vertices
But it seems not to have been satisfactorily answered.
Update:
At least in MATLAB I have been able to find the $\alpha _i$ via minimization procedure using the function fmincon and the objective function to minimize being the variance of the $\alpha _i$. An analytic procedure yet eludes me.