Defect (geometry) - A Potential Error

A Potential Error

It is tempting to think that every non-convex polyhedron has some vertices whose defect is negative. Here is a counterexample. Consider a cube where one face is replaced by a square pyramid: this elongated square pyramid is convex and the defects at each vertex are each positive. Now consider the same cube where the square pyramid goes into the cube: this is concave, but the defects remain the same and so are all positive.

Negative defect indicates that the vertex resembles a saddle point, whereas positive defect indicates that the vertex resembles a local maximum or minimum.