Introduction

The central object of this book is the measure of geometric quantities describing a subset of the Euclidean space (E^N,< ., . >), endowed with its standard scalar product.

Let us state precisely what we mean by a geometric quantity. Consider a subset S of points of the N-dimensional Euclidean space E^N, endowed with its standard scalar product < ., . >. Let G₀ be the group of rigid motions of E^N. We say that a quantity Q(S) associated to S is geometric with respect to G₀ if the corresponding quantity Q[g(S)] associated to g(S) equals Q(S), for all g $\in$ G₀. For instance, the diameter of S and the area of the convex hull of S are quantities geometric with respect to G₀. But the distance from the origin O to the closest point of S is not, since it is not invariant under translations of S. It is important to point out that the property of being geometric depends on the chosen group. For instance, if G₁ is the group of projective transformations of E^N, then the property of S being a circle is geometric for G₀ but not for G₁, while the property of being a conic or a straight line is geometric for both G₀ and G₁. This point of view may be generalized to any subset S of any vector space E endowed with a group G acting on it.

In this book, we only consider the group of rigid motions, which seems to be the simplest and the most useful one for our purpose. But it is clear that other interesting studies have been done in the past and will be done in the future, with different groups, such as the affine group (see [23, 36]), projective group, quaternionic group, etc.