We obtain characterizations of nearly strong convexity and nearly very convexity by using the dual concept of S and WS points, related to the so-called Rolewicz's property (α). We give a characterization of those points in terms of continuity properties of the identity mapping. The connection between these two geometric properties is established, and finally an application to approximative compactness is given.
Zihou ZHANG
,
Vicente MONTESINOS
,
Chunyan LIU
. SOME METRIC AND TOPOLOGICAL PROPERTIES OF NEARLY STRONGLY AND NEARLY VERY CONVEX SPACES[J]. Acta mathematica scientia, Series B, 2020
, 40(2)
: 369
-378
.
DOI: 10.1007/s10473-020-0205-7
[1] Bandyopadhyay P, Li Y J, Lin B L, Narayana D. Proximinality in Banach space. J Math Anal Appl, 2008, 341:309-317
[2] Diestel J. Sequences and Series in Banach Spaces. GTM 92. Springer, 1984
[3] Szymon Draga. On weakly uniformly rotund norms which are not locally uniformly rotund. Colloquium Mathematicum, 2015, 138(2):241-246
[4] Efimov N W, Stechkin S B. Approximate compactness and Chebyshev sets. Sorit Mathematics, 1961, 2:1226-1228
[5] Fabian M, Habala P, Hájek P, Montesinos V, Zizler V. Banach Space Theory; The Basis for Linear and NonLinear Analysis. Springer, 2011
[6] Fang X N, Wang J H. Convexity and Continuity of the metric projection. Math Appl, 2001, 14(1):47-51(in Chinesse)
[7] Giles J R, Gregory D A, Sims B. Geometrical implications of upper semi-continuity of the duality mapping on Banach space. Pacific J Math, 1978, 79(1):99-109
[8] Guirao A J, Montesinos V. A note in approximative compactness and continuity of the metric projection in Banach spaces. J Convex Anal, 2011, 18:341-397
[9] Hájek P, Montesinos V, Zizler V. Geometry and Gâteaux smoothness in separable Banach spaces. Operators and Matrices, 2012, 6(2):201-232
[10] Hudzik H, Kowalewski W, Lewicki G. Approximative compactness and full rotundity in Musielak-Orlicz space and Lorentz-Orlice spaces. Z Anal Anwen Aungen, 2006, 25:163-192
[11] Moltó A, Orihuela J, Troyanski S, Valdivia M. A Nonlinear Transfer Technique for Renorming//Lecture Notes in Math 1951. Berlin:Springer, 2009
[12] Montesinos V. Drop property equals reflexivity. Studia Mathm, 1987, 87(1):93-100
[13] Wang J H, Zhang Z H. Characterization of the property (C-κ). Acta Math Sci Ser A Chinese Ed, 1997, 17A(3):280-284
[14] Zhang Z H, Liu C Y. Convexity and existence of the farthest point. Abstract and Applied Analysis 2011. Article ID 139597. Doi:10.1155/2011/139597, 2011
[15] Zhang Z H, Liu C Y. Convexity and proximinality in Banach space. J Funct Spaces Appl Art, ID 724120, 2012
[16] Zhang Z H, Montesinos V, Liu C Y, Gong W Z. Geometric properties and continuity of pre-duality mapping in Banach spaces. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales Serie A, Matemáticas, 2015, 109(2):407-416
[17] Zhang Z H, Shi Z R. Convexities and approximative compactness and continuity of metric projection in Banach spaces. J Approx Theory, 2009, 2(161):800-812
[18] Zhang Z H, Zhou Y, Liu C Y. Near convexity, near smoothness and approximative compactness of half space in Banach space. Acta Math Sinica (Engl Ser), 2016, 32(5):599-606
[19] Liu C Y, Zhang Z H, Zhou Y. A note approximative compactness and midpoint locally K-uniform rotundity in Banach spaces. Acta Mathematica Scientia, 2018, 38B(2):643-650
