Abstract: In this paper, we give some characteristic properties of star-shaped sets which include a subset of a convex metric space.Using the characteristic properties, we discuss the existence problems of fixed points ofnonexpansive type mappings on star-shaped subsets of, convex metric spaces, which generalize the recent results obtained by Ding Xie-ping, Beg and Azam.Finally, we give an example which shows that our generalizations are essential.

Key words: star-shaped set, star centre, normal structure of a star-shaped set