On quasi-denting points, denting faces and the geometry of the unit ball of \(d(w,1)\)
Some methods to characterize the quasi-denting points of a closed convex bounded set \(C\) in a Banach space are discussed. This property of points has been recently introduced by R. Giles and W. B. Moors, and its relevance increased by the fact (proved by the third author of the paper) that a Banach space such that every point of the unit sphere is quasi- denting can be renormed to be LUR. It is known that the denting points of \(C\) are its extreme points of continuity, nevertheless in the paper it is shown that a similar statement does not hold for quasi-denting points. However the characterization of denting points through Bochner integrable functions is extended to quasi-denting points. The concept of denting face introduced in this paper, provides another approach to check this property, since quasi-denting points of such faces are quasi-denting for the whole set \(C\). The authors show that every face with the Krein-Milman property, made up by points of continuity, is denting. From these results they deduce that all faces of the unit sphere \(S\) of the Lorentz sequence spaces \(d(w,1)\) are denting, and all the points of \(S\) are quasi-denting for the unit ball of \(d(w,1)\).
quasi-denting points of a closed convex bounded set
scientific article; zbMATH DE number 687085