is independent of the dimension of the ambient space $n$ and is the same for all $n \ge l+1$;

2.  \[reduce\] $\delta(\mathbf{A}, \mathbf{B}) $ reduces to the Grassmann distance between $\mathbf{A}$ and $\mathbf{B}$ when $k = l$;

3.  \[explicit\] $\delta(\mathbf{A}, \mathbf{B}) $ may be computed explicitly as $$\label{eq:grassdist2}
    \delta(\mathbf{A},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^{\min \{k,l\}}\theta_i(\mathbf{A},\mathbf{B})^2 \Bigr)^{1/2}$$ where $\theta_i(\mathbf{A},\mathbf{B})$ is the $i$th principal angle between $\mathbf{A}$ and $\mathbf{B}$, $i =1,\dots,\min(k,l)$.

Rewriting as $$\min_{\mathbf{X}\in \Omega_{+}(\mathbf{A})}d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B}) = \min_{\mathbf{Y} \in \Omega_{-}(\mathbf{B})} d_{\operatorname{Gr}(k,n)}(\mathbf{Y},\mathbf{A}),$$ the equation says that the distance from $\mathbf{B}$ to the nearest $l$-dimensional subspace that contains $\mathbf{A}$ equals the distance from $\mathbf{A}$ to the nearest $k$-dimensional subspace contained in $\mathbf{B}$. This relation has several parallels. We will see that:

1.  the Grassmann distance may be replaced by any of the distances in Table \[tab:distances\] (see Theorem \[thm:othermetrics\]);

2.  ‘nearest’ may be replaced by ‘furthest’ and ‘min’ above replaced by ‘max’ when $n$ is sufficiently large (see Proposition \[prop:prob4\]);

3.  ‘distance’ may be replaced by ‘volume’ with respect to the intrinsic uniform probability density on the Grassmannian (see Section \[sec:volume\]).

We will prove Theorem \[thm1\] by way of the next two lemmas.

\[lem:ineq1\] Let $k \le l \le n$ be positive integers. Let $\delta :\operatorname{Gr}(k,n)\times \operatorname{Gr}(l,n)\to [0,\infty)$ be the 