Is there an algorithm to determine which face of an n-dimensional hypercube is closest to a given point in $O(n\log(n))$?
After a bit more thinking, I realised it's actually doable in $O(n)$

If you normalise your hypercube onto the origin then to find the face the point lies on you simply need to find the axis of the point with the largest magnitude.

So, if the point is at $(0.2, 0.4, 0.7, 0.5)$, the point will be closest to the face facing in the positive z direction.