Context and Notation

While other spaces have been considered, most of the theory concerns subsets of \(n\)-dimensional Euclidean space R\(^n\) and this is the setting here. The origin in R\(^n\) is denoted by \(o \) and \(S^{n-1}\) is the unit sphere. If \(x\) ∈ R\(^n\setminus\{o\}\), then \(x^{\perp}\) is the hyperplane through the origin orthogonal to \(x\). If \(S\) is a subspace (always meaning linear subspace) of R\(^n\) and \(E\) is a subset of R\(^n\), then \(E|S\) is the projection (always meaning orthogonal projection) of \(E\) onto \(S\).