An Analysis of the Measure of [A]l in Geometric Probability

Abstract

One familiar concept from geometry is that of flat spaces such as lines and planes. By analogy, we can consider higher-dimensional analogues, as well as the 0-dimensional equivalent, which is easily shown to be a point.1 An n-dimensional space of this type is symbolized as Rn ; since all such spaces are identical for a given n, it is treated as a single object rather than a category. Extending the analogy, just as we can consider a line drawn through the origin of a 2-dimensional graph, we can consider a copy of R k going through the origin of Rn (for k<n). We can similarly consider the set of all such copies of Rk for a given k and n. Such a copy of Rk going through the origin of Rn is known as a linear subspace of Rn , and the set of all such linear subspaces is known as a Grassmanian and denoted Gr(n,k). These linear subspaces can in turn have lower-dimensional subspaces of their own; the resultant structure shows certain similarities to the structure of subsets of a given set .