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Journal of Convex Analysis 11 (2004), No. 2, 477--494
Copyright Heldermann Verlag 2004

Kuratowski's Index of Non-Compactness and Renorming in Banach Spaces

F. Garcia
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

A point x in A, where A is a subset of the metric space (X, || . ||), 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.

Keywords: Quasi-denting points, Kuratowski's index, LUR renorming.

MSC 2000: 46B03, 46B20.

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