Why Is a Milk Carton Considered a Nonconvex Polyhedron?

[JasonJo]

Why is a Milk Carton an example of a nonconvex polyhedron such that the sum of the face angles around each vertex is no more than 360 degrees.
- I don't know how a milk carton qualifies as a nonconvex polyhedron. isn't every face a convex polygon?

Is it possible to have a polyhedron with 7 edges? Why or why not?
- I suspect it's not possible, but I can't 100% justify why not. I tried arguing that it's impossible to have a polyhedron with 7 edges because every face must be a polygon, but i don't know how to finish that statement.

Given a vertice v and a winged edge data structure, describe how to create a sorted list of all edges incident to v.
- ?

 

[HallsofIvy]

Yes, every face is a convex polygon- that's not relevant to the carton itself being a non-convex polyhedron. Look at the top, where you open it to form a spout- that parts goes into the polyhedron so it is not convex.

Do you know Euler's formula? For a simply connected (no holes through it) polyhedron, the number of faces, minus the number of edges, plus the number of vertices = 2. The simplest polyhedron, a tetrahedron, has 4 faces, 6 edges, 4 vertices: 4- 6+ 4= 2. If you add one more edge, for Euler's formula to stay true, you would have to add either a vertex or a face but not both! Can you do that?