, Volume 86, Issue 1-3, pp 81-91

On Jordan Angles and the Triangle Inequality in Grassmann Manifolds

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Abstract

Let L, M and N be p-dimensional subspaces in \(\mathbb{R}\) n . Let {ψ j } be the angles between L and M, let {ψ j } be the angles between M and N, and let {θ j } be the angles between L and M. Consider the orbit of the vector ψ = (ψ1,...., ψ n ) ∈ \(\mathbb{R}\) p with respect to permutations of coordinates and inversions of axes. Let Z be the convex hull of this orbit. Then θ ∈ ϕ + Z. We discuss similar theorems for other symmetric spaces. We also obtain formula for geodesic distance for arbitrary invariant convex Finsler metrics on classical symmetric spaces.