This Demonstration shows that the intersection of an arbitrary convex polyhedron by an arbitrary plane is a convex polygon.

The polyhedron is created by randomly selecting points and then taking the convex hull of those points. It is possible that the actual number of vertices will be less than the chosen number of points.

[1] M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars, Computational Geometry Algorithms and Applications, 3rd ed., Berlin: Springer, 2008 pp. 243–258.