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.

This monograph is the first in a projected series on Probability Theory. Though its title “Integral Geometry” may appear somewhat unusual in this context it is nevertheless quite appropriate, for Integral Geometry is an outgrowth of what in the olden days was referred to as “geometric probabilities.”

Originating, as legend has it, with the Buffon needle problem (which after nearly two centuries has lost little of its elegance and appeal), geometric probabilities have run into difficulties culminating in the paradoxes of Bertrand which threatened the fledgling field with banishment from the home of Mathematics. In rescuing it from this fate, Poincare made the suggestion that the arbitrariness of definition underlying the paradoxes could be removed by tying closer the definition of probability with a geometric group of which it would have to be an invariant.

Thus a union of concepts was born that was to become Integral Geometry.

It is unfortunate that in the past forty or so years during which Probability Theory experienced its most spectacular rise to mathematical prominence, Integral Geometry has stayed on its fringes. Only quite recently has there been a reawakening of interest among practitioners of Probability Theory in this beautiful and fascinating branch of Mathematics, and thus the book by Professor Santaló, for many years the undisputed leader in the field of Integral Geometry, comes at a most appropriate time.