Consider the unit hypercube in $\mathbb{R}^N$.
$\mathcal{P} = \{\mathbf{x} ~| ~x_i \in [0,1] \text{ for } i=1,\ldots,N \}$
and a half-space which intersects the unit cube:
$\mathcal{H} = \{\mathbf{x} ~| ~a^T \mathbf{x} \geq 0 \}$
I am wondering: if we know that $\mathcal{P} \cap \mathcal{H} \neq \emptyset$, then is there a simple way to determine the center of gravity of $\mathcal{P} \cap \mathcal{H}$?
The center of gravity of the intersection is a weighted sum of the centers of gravity of the uncut cube and of the portion that was cut off, with a weighting factor that depends on the relative volumes of these two regions. The main difficulty will be finding the volume of the cut-off portion, which is in general of exponential complexity in $N$ when you only have it defined by its facets rather than vertices. You are helped by the fact that it is a convex region.