Abstract

We prove that if every bounded subset of X∗X^{*} is w∗w^{*}-separable, XX is compactly locally uniformly convex, XX is 2-strictly convex and XX is nonsquare, then there exists a sequence {xn}n=1∞\{x_n\}_{n = 1}^\infty of dentable points of B(X)B(X) such that S(X)⊂∪n=1∞B(xn,rn)S(X) \subset \mathop \cup _{n = 1}^\infty B(x_n,{r_n}), where rn<1r_{n}< 1 for all n∈Nn\in N. Moreover, we also prove that if AA is a bounded closed convex subset of XX, then x∈Ax\in A is a strongly exposed point of AA if and only if xx is a dentable point of AA and xx is a w∗w^{*}-exposed point of Aw∗‾\overline {{A^{{w^*}}}}.

Author information

Contact details are reproduced from the original publication and may be historical.

Shaoqiang Shang

Dept. of Mathematics, Northeast Forestry University, Harbin 150040, P. R. China

sqshang@163.com

Yunan Cui

Dept. of Mathematics, Harbin University of Science and Technology, Harbin 150080, P. R. China

cuiya@hrbust.edu.cn

Keywords

  • Compactly locally uniformly convex
  • ball-covering property
  • dentable point
  • nonsquare space
  • 2-strictly convex space

Mathematics Subject Classification

46B20

References

  1. S. Chen, H. Hudzik, W. Kowalewski, Y. Wang, M. Wisla: Approximative compactness and continuity of metric projector in Banach spaces and applications, Sci. China Ser. A 50(2) (2007) 75–84.https://doi.org/10.1007/s11425-007-0142-0Google Scholar
  2. L. Cheng: Ball-covering property of Banach spaces, Israel J. Math. 156(1) (2006) 111–123.https://doi.org/10.1007/BF02773827Google Scholar
  3. L. Cheng: Erratum to: Ball-covering property of Banach spaces, Israel J. Math. 184(1) (2011) 505–507.https://doi.org/10.1007/s11856-011-0078-5Google Scholar
  4. L. Cheng, Q. Cheng, X. Liu: Ball-covering property of Banach spaces is not preserved under linear isomorphisms, Sci. China Ser. A 51(1) (2008) 143–147.https://doi.org/10.1007/s11425-007-0102-8Google Scholar
  5. L. Cheng, Q. Cheng, H. Shi: Minimal ball-covering in Banach spaces and their application, Studia Math. 192(1) (2009) 15–27.https://doi.org/10.4064/sm192-1-2Google Scholar
  6. L. Cheng, Z. Luo, X. Liu: Several remarks on ball-covering property of normed spaces, Acta Math. Sin. 26(9) (2010) 1667–1672.https://doi.org/10.1007/s10114-010-9036-0Google Scholar
  7. L. Cheng, H. Shi, W. Zhang: Every Banach spaces with a w∗ -separable dual has an 1 + ε-equivalent norm with the ball covering property, Sci. China Ser. A 52(9) (2009) 1869–1874.https://doi.org/10.1007/s11425-009-0175-7Google Scholar
  8. V. P. Fonf, C. Zanco: Covering spheres of Banach spaces by balls, Math. Ann. 344(4) (2009) 939–945.https://doi.org/10.1007/s00208-009-0336-6Google Scholar
  9. B. Lin, P. Lin, S. Troyanski: Characterizations of denting points, Proc. Amer. Math. Society 102(3) (1988) 526–528.https://doi.org/10.1090/S0002-9939-1988-0928972-1Google Scholar
  10. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, Lecture Notes Math. 1364, Springer, New York (1989).https://doi.org/10.1007/978-3-662-21569-2Google Scholar
  11. S. Shang, Y. Cui: Ball-covering property in uniformly non-l3 (1) Banach spaces and application, Abstr. Appl. Anal. 2013(1) (2013) 1–7.https://doi.org/10.1155/2013/873943Google Scholar
  12. S. Shang, Y. Cui: Locally 2-uniform convexity and ball-covering property in Banach space, Banach J. Math. Anal. 9(1) (2015) 42–53.https://doi.org/10.15352/bjma/09-1-4Google Scholar
  13. S. Shang, Y. Cui, Y. Fu: P-convexity of Orlicz-Bochner function spaces endowed with the Orlicz norm, Nonlinear Analysis 74(1) (2012) 371–379.https://doi.org/10.1016/j.na.2011.08.040Google Scholar
  14. S. Shang, Y. Cui, Y. Fu: Smoothness and approximative compactness in Orlicz function spaces, Banach J. Math. Anal. 8(1) (2014) 26–38.https://doi.org/10.15352/bjma/1381782084Google Scholar
  15. I. Singer: On the set of best approximation of an element in a normed linear space, Rev. Roumaine Math. Pures Appl. 5(1) (1960) 383–402.Google Scholar

Suggested citation

S. Shang, Y. Cui. “Dentable Point and Ball-Covering Property in Banach Spaces.” Journal of Convex Analysis 25 (2018), No. 3, 1045–1058. https://doi.org/10.68381/jca25067

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2018. Rights now held by Banach Press.