Definitions and Fundamental Formula

A set of points in the plane is called a domain if it is open and connected. A set of points is called a region if it is the union of a domain with some, none, or all its boundary points.

By a lattice of fundamental regions in the plane we understand a sequence of congruent regions α1, α1… that satisfies the following conditions:

1. Every point P of the plane belongs to one and only one region αi;

2. Every αi; can be superposed on α0 by a motion ti that superposes on every αnan αs, that is, by a motion that takes the whole lattice onto itself.

The set of motions {ti} such that α0 = tiαi is a discrete subgroup of the

group of motions. Such groups are called crystallographic groups. There are seventeen classes of nonisomorphic crystallographic groups, but for any given group there are infinitely many possible fundamental regions. It is not our purpose to present details on these groups, which are explored, for instance, in the books of Coxeter [127] and Guggenheimer [254]. Figures 8.1 to 8.5 are examples of lattices whose fundamental regions are squares, parallelograms, hexagons, or figures of more complicated shape.

Let D0 be a figure in the plane, that is, a set of points, which can be a region bounded by a finite number of closed curves without double points, a set of rectifiable curves, a finite number of points, etc. Suppose that D0 is contained in the fundamental region α1 of a given lattice.

During the years 1935–1939, W. Blaschke and his school in the Mathematics Seminar of the University of Hamburg initiated a series of papers under the generic title “Integral Geometry.” Most of the problems treated had their roots in the classical theory of geometric probability and one of the project's main purposes was to investigate whether these probabilistic ideas could be fruitfully applied to obtain results of geometric interest, particularly in the fields of convex bodies and differential geometry in the large. The contents of this early work were included in Blaschke's book Vorlesungen iiber Integralgeometrie [51].

To apply the idea of probability to random elements that are geometric objects (such as points, lines, geodesies, congruent sets, motions, or affinities), it is necessary, first, to define a measure for such sets of elements. Then, the evaluation of this measure for specific sets sometimes leads to remarkable consequences of a purely geometric character, in which the idea of probability turns out to be accidental. The definition of such a measure depends on the geometry with which we are dealing. According to Klein's famous Erlangen Program (1872), the criterion that distinguishes one geometry from another is the group of transformations under which the propositions remain valid. Thus, for the purposes of integral geometry, it seems natural to choose the measure in such a way that it remains invariant under the corresponding group of transformations. This sequence of underlying mathematical concepts – probability, measure, groups, and geometry – forms the basis of integral geometry.

Introduction

Convex sets play an important role in integral geometry. For this reason we will review here their principal properties, especially those which will be needed in the following sections. In this chapter we consider convex sets in the plane. For convex sets in n-dimensional euclidean space, see Chapter 13. For a more complete treatment, refer to the classical books of Blaschke [50] and Bonnesen and Fenchel [63], or to the more modern texts of Benson [27], Eggleston [162], Grünbaum [247], Jaglom and Boltjanski [320], Hadwiger [270], Hadwiger and coauthors [282], and Valentine [683].

A set of points K in the plane is called convex if for each pair of points A ∈ K, B ∈ K it is true that AB ⊂K, where AB is the line segment joining A and B. For convenience we shall assume throughout that the convex sets are bounded and closed.

A curve with end points P, Q, is called convex if its point set, together with the segment PQ, bounds a convex set. If the convex set K is bounded and has interior points, then the boundary of K is called a closed convex curve. Throughout, we will denote by ∂K the boundary of the set K. If all the points of K belong to ∂K, then K is a line segment.

We can prove that (a) All convex curves are piecewise differentiable (i.e., they are the union of a countable set of arcs with continuously turning tangent); in other words, convex curves have at most a countable set of corners; (b) All bounded convex curves are rectifiable.

A large body of mathematics consists of facts that can be presented and described much like any other natural phenomenon. These facts, at times explicitly brought out as theorems, at other times concealed within a proof, make up most of the applications of mathematics, and are the most likely to survive changes of style and of interest.

This ENCYCLOPEDIA will attempt to present the factual body of all mathematics. Clarity of exposition, accessibility to the non-specialist, and a thorough bibliography are required of each author. Volumes will appear in no particular order, but will be organized into sections, each one comprising a recognizable branch of present-day mathematics. Numbers of volumes and sections will be reconsidered as times and needs change.

It is hoped that this enterprise will make mathematics more widely used where it is needed, and more accessible in fields in which it can be applied but where it has not yet penetrated because of insufficient information.