Orthogonal projection of a unit cube in $R^n$ onto an ($n-1$)-dimensional subspace

Question

I am looking for a way to compute the orthogonal projection of a unit cube in $\mathbf{R}^n$ onto an ($n-1$)-dimensional subspace (hyperplane). In the case $n=3$, for example, can one say that the projection of the cube is the convex hull of the projected vertices? Is there a simple characterization of the projection for $n>3$? Also, is there any book that treats these kind of problems in some depth? Discrete matematics is really not my field...