The subject of this book is closely related to and expands classical integral geometry. In its most advanced areas it merges with those topics in geometrical probability which are now known as stochastic geometry. By the application of a number of powerful yet simple new ideas, the book makes a sophisticated field accessible to readers with just a modest mathematical background.

Traditionally, integral geometry considers only finite sets of geometrical elements (lines, planes etc.) and measures in the spaces of such sets. In the spirit of the Erlangen program, these measures should be invariant with respect to an appropriate group acting in basic space – to ensure that we are still in the domain of geometry. Assume that the basic space is ℝn (as is the case in the most of this book). If the group contains translations of ℝn, then the measures in question are necessarily totally infinite and cannot be normalized to become probability measures. Yet a step towards countably infinite sets of geometrical elements changes the situation: spaces of such sets admit probability measures which are invariant and these measures are numerous.

The step from finite sets to countably infinite sets directly transfers an integral geometrician into the domain of probability. The vast field of inquiry that opens up surely deserves attention by virtue of the mathematical elegance of its problems and as a potentially rich source of models for applied sciences.

As previous examples illustrate, neither iterative sequences nor approximate fixed point sequences for nonexpansive mappings typically converge, at least in the strong sense. However, in certain instances, these sequences may converge in the weak (or weak*) topology to fixed points, or in some other way determine invariant sets which contain fixed points.

Throughout this chapter we shall, as usual, assume that K is a nonempty, closed and convex subset of a Banach space X but, in general, we shall not assume K is bounded.

Suppose T: K→K is nonexpansive and fix x0 ∈ K. We begin by considering the iterative sequence {xn} = {Tnx0}. The set of points O(x0) = {xn: n = 0, 1, …} is called the orbit of x0 under T, and its closure is called the closed orbit. Since T is nonexpansive, if O(x0) is bounded for at least one x0 ∈ K then all other orbits O(x), x ∈ K, are bounded. Thus the nonexpansive self-mappings of K fall into two categories: those with bounded orbits and those with unbounded orbits. Obviously all nonexpansive mappings having fixed points have bounded orbits. (Less obvious is the fact that in a finite dimensional Banach space O(x0) is bounded for a nonexpansive mapping T whenever {xn} has a convergent subsequence (see Roehrig and Sine, 1981).)

In our previous examples of fixed point free, nonexpansive mappings none of the interative sequences contain convergent subsequences. However Edelstein has shown (1964) that even if such convergent subsequences exist, a nonexpansive mapping may remain fixed point free.

We have already seen that some bounded, closed and convex sets K in certain Banach spaces have the property that every nonexpansive self-mapping T: K→K must have a fixed point. When this is the case we say that K has the fixed point property (f.p.p.) for nonexpansive mappings and, if it is clear that only nonexpansive mappings are being considered, we shall simply say that K has the fixed point property or that K has f.p.p. Also, unless otherwise specified, we shall always assume that K is nonempty, bounded, closed and convex.

The problem of determining conditions on K (or on the space X containing K) which always insure that K has the f.p.p. has its origins in four papers which appeared in 1965. In the first of these (Browder, 1965a), F. Browder proved that a bounded, closed, convex set K ⊂ X has f.p.p. if X is a Hilbert space. Almost immediately, both Browder (1965b) and Göhde (1965) proved that the same is true if X belongs to the much wider class of ‘uniformly convex’ spaces (discussed in the next chapter). At the same time Kirk (1965) observed that the presence of a geometric property called ‘normal structure’ guarantees that K ⊂ X has f.p.p. if X is reflexive. The concept of normal structure was introduced in 1948 by Brodskii and Milman (1948) to study fixed points of isometries, and it is a property shared by all uniformly convex spaces.

The term ‘Metric’ Fixed Point Theory refers to those fixed point theoretic results in which geometric conditions on the underlying spaces and/or mappings play a crucial role. Obviously there can be no clear line separating this branch of fixed point theory from either the topological or set-theoretic branches since metric methods are often useful in proving results which are basically nonmetric in nature, and vice versa. However, the results considered here are always couched in at least a metric space framework, usually in a Banach space setting, and the methods typically involve both the topological and the geometric structure of the space in conjunction with metric constraints on the behavior of the mappings.

For the past twenty-five years metric fixed point theory has been a flourishing area of research for many mathematicians. Although a substantial number of definitive results have now been discovered, a few questions lying at the heart of the theory remain open and there are many unanswered questions regarding the limits to which the theory may be extended. Some of these questions are merely tantalizing while others suggest substantial new avenues of research.

It is apparent that the theory has now reached a level of maturity appropriate to an examination of its central themes. The topics selected for this text were chosen accordingly. No attempt has been made to explore all aspects of the theory nor to present a compendium of known facts.

The aim of this chapter is to present some basic mathematical tools on which many constructions in the subsequent chapters depend.

Thus we will often refer to what we call the ‘Cavalieri principle’. We try to revive this old familiar name because of the surprising frequency with which the transformations Cavalieri considered about 350 years ago occur in integral geometry.

No less useful will be the principles which we call ‘Lebesgue factorization’ and ‘Haar factorization’. The first is a rather simple corollary of a well-known fact that in ℝn there is only one (up to a constant factor) locally-finite measure which is invariant with respect to shifts of ℝn, namely the Lebesgue measure. Haar factorization is a similar corollary of a much more general theorem of uniqueness of Haar measures on topological groups. We use the two devices in the construction of Haar measures on groups starting from Haar measures on subgroups.

Integral geometry binds together such notions as metrics, convexity and measures, and these interconnections remain significant throughout the book; §§1.7 and 1.8 are introductory to this topic.

The Cavalieri principle

The classical Cavalieri principle in two dimensions can be formulated as follows.

Let D1 and D2 be two domains in a plane (see fig. 1.1.1).

If for each value of y the length of the chords X1 and X2 coincide, then the areas of D1 and D2 are equal.