the required distance $\delta(\mathbf{A}, \mathbf{B}) $ as that between the $k$-plane $\mathbf{A}$ and the closest $k$-plane $\mathbf{Y}$ contained in $\mathbf{B}$, measured within $\operatorname{Gr}(k,n)$. In other words, we want the Grassmann distance from $\mathbf{A}$ to the closed subset $\Omega_{-}(\mathbf{B})$, $$\label{eq:minus}
\delta(\mathbf{A}, \mathbf{B}) \coloneqq d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A}, \Omega_{-}\bigl(\mathbf{B})\bigr) = \min\bigl\{ d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{Y}) :  \mathbf{Y}\in \Omega_{-}(\mathbf{B})\bigr\}.$$ This has the advantage of being intrinsic — the distance $\delta(\mathbf{A}, \mathbf{B}) $ is measured in $d_{\operatorname{Gr}(k,n)}$ and is defined wholly within $\operatorname{Gr}(k,n)$ without any embedding of $\operatorname{Gr}(k,n)$ into an arbitrary ambient space. Furthermore, by the property of $d_{\operatorname{Gr}(k,n)}$ in Corollary \[cor:infty\], $\delta(\mathbf{A}, \mathbf{B}) $ does not depend on $n$ and takes the same value for any $m \ge n$. We illustrate this in Figure \[fig:sphere\]: The sphere is intended to be a depiction of $\operatorname{Gr}(1,3)$ though to be accurate antipodal points on the sphere should be identified.

(-4,0) circle\[radius=0.05\]; at (-4.08,0) [$\mathbf{A}$]{}; (0,4) arc (90:270:2cm and 4cm); (0,4) arc (90:-90:2cm and 4cm); (0,0) circle (4cm); (0,0) circle (4cm); at (3,-3) [$\operatorname{Gr}(1,3)$]{}; at (0,2.5) [$\Omega_{-}(\mathbf{B})$]{}; at (-3,-1.25) [$\gamma$]{}; (-4,0) arc (180:243:4cm and 2cm); (-1.8,-1.8) circle\[radius=0.05\]; at (-2,-1.9) [$\mathbf{X}$]{};

However, it is equally natural to define $\delta(\mathbf{A}, \mathbf{B}) $ as the distance between the $l$-plane $\mathbf{B}$ and the closest $l$-plane $\mathbf{Y}$ 