We discuss the Alexandrov-Toponogov comparison theorem under the conditions of radial curvature of a pointed manifold $(M,o)$ with reference surface of revolution $(\widetilde M, \tilde o)$. There are two obstructions to make the comparison theorem for a triangle one of whose vertices is a base point $o$. One is the cut points of another vertex $\tilde p \not=\tilde o$ of a comparison triangle in $\widetilde M$. The other is the cut points of the base point $o$ in $M$. We find a condition under which the comparison theorem is valid for any geodesic triangle with a vertex at $o$ in $M$.

We introduce a new constant for the surfaces of revolution homeomorphic to the 2-sphere. We prove a sphere theorem for radial curvature, assuming an inequality in the constant and the ratio of the difference of the maximal distance to the base point from the diameter of the reference surface and the injectivity radius of the base point. Namely, if a compact pointed Riemannian $n$-manifold which is referred to a surface of revolution satisfies the inequality, then it is topologically an $n$-sphere.

In this paper, we study regular norms on $\mathbb{R}^n$. It is shown that the class of regular norms is a natural generalization of that of generalized Day-James type norms. Furthermore, using absolute norms, we give some characterizations of regular norms.

In this paper we introduce a new kind of pair of finite range sets in $\mathbb{C}$ for meromorphic functions corresponding to their uniqueness.