The Core of a Grassmannian Frame

Let X = { x i } i = 1 m be a set of unit vectors in R n . The coherence of X is coh ( X ) : = max i ≠ j | ⟨ x i , x j ⟩ | . A vector x ∈ X is said to be isolable if there are unit vectors x ′ arbitrarily close to x such that | ⟨ x ′ , y ⟩ | < coh ( X ) for all other vectors y in X . We define the core of a Grassmannian frame X = { x i } i = 1 m in R n at angle α as a maximal subset of X which has coherence α and has no isolable vectors. In other words, if Y is a subset of X , coh ( Y ) = α , and Y has no isolable vectors, then Y is a subset of the core. We will show that every Grassmannian frame of m > n vectors for R n has the property that each vector in the core makes the angle α with a spanning family from the core. Consequently, the core consists of ≥ n + 1 vectors. We then develop other properties of Grassmannian frames and of the core.