# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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
Full Text:
##### References:
 [1] Borwein, J. M.; Fitzpatrick, S.: Mosco convergence and the kadec property, Proc. amer. Math. soc. 106, 843-851 (1989) · Zbl 0672.46007 · doi:10.2307/2047444 [2] Brosowski, B.; Deutsch, F.; Neürnberge, G.: Parametric approximation, J. approx. Theory 29, 261-271 (1980) · Zbl 0483.41033 · doi:10.1016/0021-9045(80)90115-X [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, No. 2, 293-303 (2008) · Zbl 1153.46008 · doi:10.1007/s11425-007-0142-0 [4] Fang, X. N.; Wang, J. H.: Convexity and continuity of metric projection, Math. appl. 14, No. 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 · doi:10.1017/S0004972700018384 [6] He, R. Y.: K-strongly convex and locally K-uniformly smooth spaces, J. math. (PRC) 17, No. 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 · doi:10.4171/ZAA/1283 [8] Liu, P. D.; Hou, Y. L.: A convergence theorem of martingales in the limit, Northeast. math. J. 6, No. 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, No. 6, 1521-1524 (1972) · Zbl 0268.46017 [10] Singer, I.: The theory of best approximation and functional analysis, (1974) · Zbl 0291.41020 [11] Sullivan, F.: Geometrical properties determined by the higher dual of a Banach space, Illinois J. Math. 21, 315-381 (1977) · Zbl 0363.46024 [12] Tsukada, M.: Convergence of best approximations in a smooth Banach space, J. approx. Theory 40, 301-309 (1984) · Zbl 0545.41042 · doi:10.1016/0021-9045(84)90003-0 [13] Wang, H. J.: Convergence theorems for best approximations in a nonreflexive Banach space, J. approx. Theory 93, No. 3, 480-490 (1998) · Zbl 0924.41021 · doi:10.1006/jath.1998.3170 [14] Wang, H. J.: Some results on the continuity of metric projections, Math. appl. 8, No. 1, 80-84 (1995) · Zbl 0949.46501 [15] Wang, H. J.: The metric projections in nonreflexive Banach space, Acta math. Sci. 26, No. A, 840-846 (2006) · Zbl 1116.46301 [16] Wang, H. J.; Nan, C. X.: On the convergence of $\varepsilon$-approximation, (1996) [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), No. 3, 280-284 (1997) · Zbl 0917.46013 [19] Wu, C. X.; Li, Y. J.: Strong convexity in Banach space, J. math. (PRC) 13, No. 1, 105-108 (1993) · Zbl 0802.46026