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?