Measures of noncompactness and related properties

Attempts to classify properties of the ball B, or the space X, utilizing the notion of measures of noncompactness are presented. They are connected with the Kadec–Klee property. Measures of noncompactness are used to generalize the notion of uniform convexity and smoothness.

Smoothness and uniform smoothness

The notions of smoothness and uniform smoothness of a space are discussed. The relation with differentiability of the norm is shown. The main tool, the modulus of smoothness of a space is studied.

Three special topics

First, the girth of the sphere, the infimum of length of arcs joining antipodal points on the sphere is discussed. Second, a coefficient measuring maximal separation of sequences contained in the ball is presented. Special properties of weakly convergent sequences are observed.

More moduli and coefficients

The general construction of multi-dimensional Milman’s moduli is described. Two-dimensional moduli are related to uniform convexity and uniform smoothness. The James constant measuring nonsquareness of the ball is discussed. A universal modulus, called also the modulus of squareness, and related both to convexity and smoothness is studied.

Strict and uniform convexity

Ways to classify and measure convexity of balls are described. Properties like strict convexity, uniform convexity, and squareness are discussed. The main tool, the modulus of convexity of a space, is studied. In the case of uniformly convex spaces, nearest point projections and asymptotic centres of sequences are presented.

Low dimensional spaces

The chapter is devoted to build the reader’s geometrical intuition. It contains a list of examples of norms in low dimensional spaces. Special features of two-dimensional spaces are described in more detail.

Basics and prerequisites

The chapter contains notation and an overview of prerequisites needed to understand the further text. They mostly correspond to the standard course of functional analysis. A few more advanced subjects like criteria of reflexivity, finite dimensional decompositions, etc., are briefly described.

The case of Banach lattices

Uniform monotonicity and order uniform smoothness for Banach lattices are discussed as counterparts of uniform convexity and uniform smoothness. Corresponding moduli are defined. Analogies and differences are presented.

Radius vs diameter

The subject of the chapter is the relationship between the (Chebyshev) radius and diameter of convex bounded sets. The main tool is the Jung coefficient. Diametral sets and normal structure in connection with the fixed point theory for nonexpansive mappings are presented.

Projections on balls and convex sets

Radial projection on the unit ball and its Lipschitz constant are discussed. Special attention is paid to nonexpansive projections on balls and other sets. The cases of Hilbert spaces and spaces with uniform norm are studied in more detail.