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In this paper, we investigate the continuities of the metric projection in a nonreflexive Banach space X , which improve the results in [X.N. Fang, J.H. Wang, Convexity and continuity of metric projection, Math. Appl. 14 (1) (2001) 47-51; P.D. Liu, Y.L. Hou, A convergence theorem of martingales in the limit, Northeast. Math. J. 6 (2) (1990) 227-234; H.J. Wang, Some results on the continuity of metric projections, Math. Appl. 8 (1) (1995) [80][81][82][83][84]. Under the assumption that X has some convexities, we discuss the relationship between approximative compactness of a subset A of X and continuity of the metric projection P A . We also give a representation theorem for the metric projection to a hyperplane in dual space X * and discuss its continuity.

DEDICATED TO PROFESSOR KY FANOn the unit ball of an Orlicz function space the denting points, weak U denting points, and quasi-denting points coincide. But on the unit ball of an Orlicz sequence space the quasi-denting points are different from the denting points and weak U denting points. We also show that the weak U drop property and the Kadec᎐Klee property are equivalent in Orlicz spaces. Hence the weak drop property and the weak U drop property are independent properties in Banach spaces. ᮊ

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