Before revision:   "Nearly flat" means that every  face normal forms a sufficiently small angle  with the z-axis . Although the result is not surprising, the proof relies on some recently developed concepts , angle-monotone and radially monotone curves  .
After revision:  "Nearly flat" means that every face normal forms a sufficiently small angle with the z-axis orthogonal to the halfspace bounding plane. The size of Phi depends on the acuteness gap alpha: if every triangle angle is at most pi/2-alpha, then Phi ~= 0.36 sqrt(alpha) suffices; e.g., for alpha ~= 3deg, Phi = 5deg. Even if C is closed to a polyhedron by adding the convex polygonal base under C, this polyhedron can be edge-unfolded without overlap. The proof employs the recent concepts of angle-monotone and radially monotone curves .