We work over $\mathbb{R}^N$. Let $\mathbf{P}\_1$ denote the hyperplane constructed using $N$ points, each of which is on a different axis (there are $N$ axes). We denote by $\mathbf{P}\_2$ the convex hull of a set of $M$ points, but we don't know which of these points are the vertices.
I am trying to find the minimum distance between $\mathbf{P}\_1$, which can also be seen as a convex hull, and $\mathbf{P}\_2$.


My approach is to define an optimization problem having an acceptable computational complexity. My question is how can we define such a problem ? 



Find the minimum distance between two convex hulls
This is discussed at length [here.](http://www.researchgate.net/publication/224108603_A_Fast_Geometric_Algorithm_for_Finding_the_Minimum_Distance_Between_Two_Convex_Hulls) (Kaown and Liu, 2009)


