Proof for sum of vertex polygons in polyhedra

Question

What's the rigirous proof for the statement:

When the internal angles meeting at a vertex are added, if the sum $<360$ then it's the polyhedra is convex, if the sum $= 360$ it's flat and $>360$ is concave.

Answer 1

The statement you gave, being out of context, clearly is wrong.

Even when considering the additional restriction to uniform polyhedra only, then there still exist counterexamples for your proposition: e.g. the small stellated dodecahedron (one of the Kepler-Poinsot solids) has 5 regular pentagrams per vertex, i.e. it has a local sum of $\,5\cdot 36°=180°<360°$, but because of using pentagrams it clearly is a (globally) non-convex (aka concave) polyhedron.

Btw., would you think that this magic would occur only because of the starry facial pentagrams? Well then consider a usual regular icosahedron with the top-most pentagonal pyramidal cap dimpled inwards instead. Then the such deformed polyhedron obviously becomes concave, while the local vertex sum still remains $\,5\cdot 60°=300°<360°\,$ all over on any of its vertices!

--- rk