Volume of metric balls in real Grassmann manifold with an application to frames

Volume estimates of metric balls in manifolds find diverse applications in various applied sciences including information theory and coding theory. In this paper, the focus is on the exact volume of metric balls in real Grassmann manifold, extending recent results of the complex case. Firstly, we formulate the exact volume as a multi-dimensional integral, which is then represented as an inverse Laplace transform of a Pfaffian. The Pfaffian structure arises as a result of the de-Bruijn formula for multi-dimensional integrals. Using the exact representation, examples of explicit volume formulas are given. As an application, we study the performance bound of the worst-case coherence of Grassmannian frames.