Forming a polyhedron from concave polygonal faces.

Question

A polyhedron is a convex, three dimensional region bounded by a finite number of polygonal faces.

So is it possible that some of those polygonal faces be concave ? Can concave polygons be used in the process to form a 3D convex region ?

Answer 1

No. Every intersection of two convex shapes (such as a convex polyhedron and the plane through one of its faces) is convex.

Answer 2

If you allow two faces to be coplanar (which would be an unusual admission!), then you could build a cube of six copies each of the triangle and nonconvex pentagon shown left below.

But if no two faces are coplanar, then in the neighborhood of a nonconvex point on the boundary of a nonconvex face, there must be two points determining a segment (dashed right above) with the segment exterior to the polyhedron. And this means the polyhedron fails the definition of convexity, and so is itself nonconvex.