Intrinsic characterization of a star shaped domain Let A be a closed (compact no boundary), embedded (no self intersections), smooth surface of R^3. We say that the interior of A is star shaped if there exists a point p in A, such that for any point q in A, the line segment joining p and q lies entirely within A. My questions is this:
Let (S^2,g) be the sphere with a given smooth Riemannian metric (may not be the constant curvature one), and assume there exists a smooth isometric embedding f: S^2 -> R^3. Does there exist a condition on the Gaussian curvature of g that ensures the interior of f(S^2) (the image of S^2 under f) is star shaped? 
 A: Positive Gauss curvature certainly suffices, by the theorem of Cohn-Vossen, which implies that the embedded surface is convex. Beyond that, I'm not aware of any other intrinsic characterization of the boundary of a star-shaped domain. It sounds quite difficult to me, since neither existence nor uniqueness of the isometric embedding is known, if the curvature is not everywhere non-negative (positive?).
A: Here is an example of a smooth sphere in R^3 that is not star shaped, and such that both positive and negative cuvature parts are connected. Take in R^3 a thin rotation-invariant torus. Its positive and negative curvature parts are long thin annuli. Now cut a half of it by a plane going through the axe of rotation. Take one of the halthes (a long, thin, curved tube) and cup its two little holes with two little half-spheres -- so you get a sort of banana. This can be done in a smooth way so that positive curvature part is connected.  
A: This seems very related to the Minkowski problem and its converse which you can read about in this nice survey by Herman Gluck.
I'll see if I can find more recent results and post them here.
I haven't read this book by Alexandrov, but it ought to be informative as well: Intrinsic Geometry of Convex Surfaces
