Characterisation of linearly separable points of a hypercube

Question

Essentially, linearly separable points are just those corners that can be cut off with just one slice as marked out by a hyperplane.

E.g. for a cube, the following 4 points (red) are not linearly separable - no single cut by a plane (tilted at whatever angle) across the cube can slice off exactly these 4 points:

So this begs the question: given $n$ points on an $m$-dimensional hypercube, how can I tell if these $n$ points are linearly separable?

Answer 1

This is a crucial problem in machine learning, much studied since the 60's and there is no easy characterisation or criterion - nor even efficient algorithms. See eg this and this, , and references, or google for "Threshold Logic".