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

A point x in A, where A is a subset of the metric space

, 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.