The aim of this paper is to derive some relationships between the concepts of the property of strong $(\alpha ')$ introduced recently by Hong-Kun Xu and the so-called characteristic of near convexity defined by Goebel and S\c ekowski. Particularly we provide very simple proof of a result obtained by Hong-Kun Xu.
@article{118811, author = {Wagdy Gomaa El-Sayed and Krzysztof Fraczek}, title = {Remarks on some properties in the geometric theory of Banach spaces}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {37}, year = {1996}, pages = {17-22}, zbl = {0852.47025}, mrnumber = {1396159}, language = {en}, url = {http://dml.mathdoc.fr/item/118811} }
El-Sayed, Wagdy Gomaa; Fraczek, Krzysztof. Remarks on some properties in the geometric theory of Banach spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) pp. 17-22. http://gdmltest.u-ga.fr/item/118811/
On drop property and nearly uniformly smooth Banach spaces, Nonlinear Analysis T.M.A. 14 (1990), 927-933. (1990) | MR 1058414
Compactness conditions in the geometric theory of Banach spaces, Nonlinear Analysis T.M.A. 16 (1991), 669-682. (1991) | MR 1097324
Conditions involving compactness in geometry of Banach spaces, Nonlinear T.M.A. 20 (1993), 1217-1230. (1993) | MR 1219238
Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Math., vol. 60, M. Dekker, New York, Basel, 1980. | MR 0591679
A geometric theorem useful in nonlinear analysis, Boll. Un. Mat. Ital. 6 (1972), 369-372. (1972) | MR 0317130
On densifying and related mappings and their application in nonlinear functional analysis, Theory of Nonlinear Operators, Akademie-Verlag, Berlin, 1974, pp. 15-56. | MR 0361946
A characterization of normal structure in Banach spaces, Fixed Point Theory and Applications (K.K. Tan, ed.), World Scientific, Singapore, 1992, pp. 122-129.
The modulus of noncompact convexity, Ann. Univ. Mariae Curie- Skłodowska, Sect. A 38 (1984), 41-48. (1984) | MR 0856623
Measures of noncompactness and normal type structures in Banach spaces, Panamer. Math. J. 3 (1993), 17-34. (1993) | MR 1216273 | Zbl 0846.46008
Topological Vector Spaces I, Springer Veralg, Berlin, 1969. | MR 0248498
Classical Banach Spaces, Springer Verlag, Berlin, 1973. | MR 0415253
Drop property equals reflexivity, Studia Math. 87 (1987), 93-110. (1987) | MR 0924764 | Zbl 0652.46009
On drop property, Studia Math. 85 (1987), 27-35. (1987) | MR 0879413