[blamestross]

So you won't be able to define a hull at all without (n+1) points. He has 10k points so that sounds like a lot relatively.

So the definition of a convex hull is, put generally, the set of points that define faces such that every point is on the face or on the "inside" of it (mean point side)

To test if a point is inside that simplex hull, you need to check every one of those faces. But that isn't the problem.

Every face is a "filter" that all the other points have to pass. Moving beyond the simplex, at higher dimensions, the number of faces that need to "accept" every other point scales faster in higher dimensions. The odds of that aren't horribly clear, you are right in other comments to call out that the structure of points in this context is by definition not independent or random, but you need enough structure to get around the fact that high dimensional hulls are basically all surface.