## Convexities and approximative compactness and continuity of metric projection in Banach spaces.(English)Zbl 1190.46018

The paper is concerned with the relations between various geometric properties of a Banach space $$X$$ – strong convexity, nearly strong convexity, the property of being very convex or nearly very convex – and the continuity properties of the metric projection, as upper semi-continuity (usc), strong Wijsman-Zhang usc, on closed convex subsets of $$X.$$ The relevance of the approximative compactness for the continuity of the metric projection is also emphasized. For instance, in a nearly strongly convex Banach space, a closed convex subset $$A$$ of $$X$$ is proximinal with usc metric projection $$P_A$$ iff $$A$$ is approximatively compact. The paper also contains a representation of the metric projection on $$w^*$$-closed hyperplanes in $$X^*$$ and conditions ensuring its upper semi-continuity. The obtained results improve some previous results obtained by several authors.

### MSC:

 46B20 Geometry and structure of normed linear spaces 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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### References:

 [1] Borwein, J.M.; Fitzpatrick, S., Mosco convergence and the kadec property, Proc. amer. math. soc., 106, 843-851, (1989) · Zbl 0672.46007 [2] Brosowski, B.; Deutsch, F.; Neürnberge, G., Parametric approximation, J. approx. theory, 29, 261-271, (1980) [3] Chen, S.T.; Hudzik, H.; Kowalewski, W.; Wang, Y.W.; Wisla, M., Approximative compactness and continuity of metric projector in Banach spaces and applications, Sci. China series A: mathematics, 51, 2, 293-303, (2008) · Zbl 1153.46008 [4] Fang, X.N.; Wang, J.H., Convexity and continuity of metric projection, Math. appl., 14, 1, 47-51, (2001) · Zbl 1134.41338 [5] Giles, J.R.; Sims, B.; Yorke, A.C., On the drop and weak drop properties for a Banach space, Bull austral. math. soc., 41, 503-507, (1990) · Zbl 0692.46007 [6] He, R.Y., $$K$$-strongly convex and locally $$K$$-uniformly smooth spaces, J. math. (PRC), 17, 2, 251-256, (1997) · Zbl 0936.46017 [7] Hudzik, H.; Kowalewski, W.; Lewicki, G., Approximative compactness and full rotundity in musielak – orlicz spaces and lorentz – orlicz spaces, Z. anal. anwendungen, 25, 163-192, (2006) · Zbl 1108.46016 [8] Liu, P.D.; Hou, Y.L., A convergence theorem of martingales in the limit, Northeast. math. J., 6, 2, 227-234, (1990) · Zbl 0727.60044 [9] Oshman, E.V., Characterization of subspaces with continuous metric projection into normed linear space, Soviet math., 13, 6, 1521-1524, (1972) · Zbl 0268.46017 [10] Singer, I., The theory of best approximation and functional analysis, (1974), Society for Industrial and Applied Mathematics Philadelphia [11] Sullivan, F., Geometrical properties determined by the higher dual of a Banach space, Illinois J. math., 21, 315-381, (1977) [12] Tsukada, M., Convergence of best approximations in a smooth Banach space, J. approx. theory, 40, 301-309, (1984) · Zbl 0545.41042 [13] Wang, H.J., Convergence theorems for best approximations in a nonreflexive Banach space, J. approx. theory, 93, 3, 480-490, (1998) · Zbl 0924.41021 [14] Wang, H.J., Some results on the continuity of metric projections, Math. appl., 8, 1, 80-84, (1995) [15] Wang, H.J., The metric projections in nonreflexive Banach space, Acta math. sci., 26, A, 840-846, (2006) · Zbl 1116.46301 [16] Wang, H.J.; Nan, C.X., On the convergence of $$\epsilon$$-approximation, () [17] Wang, Y.W.; Yu, J.F., The character and representative of a class of metric projection in Banach space, Acta math. sci., 21(A), 29-35, (2001) · Zbl 1018.46011 [18] Wang, H.J.; Zhang, Z.H., Characterizations of the property $$(C - \kappa)$$, Acta math. sci., 17(A), 3, 280-284, (1997) · Zbl 0917.46013 [19] Wu, C.X.; Li, Y.J., Strong convexity in Banach space, J. math. (PRC), 13, 1, 105-108, (1993) · Zbl 0802.46026
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