Abstract

A point x in A, where A is a subset of the metric space (X,∥⋅∥)(X, \|\cdot\|), is quasi-denting if for every ε > 0 there exists a slice of A containing x with Kuratowski index less than ε. The aim of this paper is to generalize the following theorem of L. S. Troyanski [Israel J. Math. 88 (1994) 175–188] with a geometric approach: A Banach space such that every point of the unit sphere is quasi-denting (for the unit ball) admits an equivalent LUR norm.

Author information

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

F. García

Dep. de Matemáticas, Fac. de Matemáticas, Universidad de Murcia, 30.100 Espinardo – Murcia, Spain

fergar@um.es

L. Oncina

Dep. de Matemáticas, Fac. de Matemáticas, Universidad de Murcia, 30.100 Espinardo – Murcia, Spain

luis@um.es

J. Orihuela

Dep. de Matemáticas, Fac. de Matemáticas, Universidad de Murcia, 30.100 Espinardo – Murcia, Spain

joseori@um.es

S. Troyanski

Inst. of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Block 8, 1113 Sofia, Bulgaria, and Dep. de Matemáticas, Fac. de Matemáticas, Universidad de Murcia, 30.100 Espinardo – Murcia, Spain

stroya@um.es

Keywords

  • Quasi-denting points
  • Kuratowski's index
  • LUR renorming

Mathematics Subject Classification

46B03, 46B20

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Suggested citation

F. García, L. Oncina, J. Orihuela, S. Troyanski. “Kuratowski's Index of Non-Compactness and Renorming in Banach Spaces.” Journal of Convex Analysis 11 (2004), No. 2, 477–494. https://doi.org/10.68381/jca11028

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Published by Heldermann Verlag, 2004. Rights now held by Banach Press LLC.