We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple (non-overlapping), planar polygon: cut along one shortest path from each vertex of P to Q, and cut all but one segment of Q.

Abstract. Consider the mapping F associating to each point x of a convex surface the set of all points at maximal intrinsic distance from x. We provide a large class of surfaces on which F is single-valued and involutive. Moreover, we show that there are point-symmetric surfaces of revolution with F single-valued but not involutive.

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We show that, in the sense of Baire categories, a typical Alexandrov surface with curvature bounded below by κ has no conical points. We use this result to prove that, on such a surface (unless it is flat), at a typical point, the lower and the upper Gaussian curvatures are equal to κ and ∞ respectively.Math. Subj. Classification (2010): 53C45