Some Geometric Properties of Quasi-Banach Spaces.
by
Fernando Albiac
Departamento de Matemáticas e Informatica. Universidad Pública de Navarra.
Coauthors: Camino Leránoz Istúriz (Universidad Pública de Navarra)

If X is a (real) Banach space, C is a closed convex subset of X, and x not in C, the convex hull of {x} \cup C is called drop. The geometric properties of drops in the particular case of C=B_X (the closed unit ball of X) are related to other geometric properties of the space such as reflexivity and uniform convexity. The concept of drop is not natural in quasi-Banach spaces because the unit ball of a quasi-Banach space does not have to be convex. If X is a quasi-Banach space, C is a closed p-convex subset of X, 0 < p < 1, and x in X\C, the p-convex hull of {x} \cup C is called p-drop. Many definitions and classical results on drops in Banach spaces can be translated in a natural way to definitions and results about p-drops in p-Banach spaces, 0 < p < 1. We study how some properties related to p-drops give us information about the p-convexity of the space. The p-Drop Property and the p-(\beta) Property are among the most interesting ones. We deal with the strict p-convexity of a quasi-Banach space in relation to the p-extreme points of its unit ball and prove that the Lorentz sequence spaces in the non-locally convex case are strictly p-convex but cannot be q-convex for any p < q <= 1. Many examples are given.

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Date received: April 6, 2000