by $d(x, S) \coloneqq \inf\{d(x,y) : y  \in S \}$. For us, $X$ is a Grassmannian, therefore compact, and so $d(x,S)$ is finite. Also, $S$ will be a closed subset and so we write $\min$ instead of $\inf$. We will introduce two possible candidates for $S$.

\[def:Omega\] Let $k,l, n \in \mathbb{N}$ be such that $k\le l \le n$. For any $\mathbf{A}\in \operatorname{Gr}(k,n)$ and $\mathbf{B}\in \operatorname{Gr}(l,n)$, we define the subsets $$\Omega_{+}(\mathbf{A})\coloneqq\bigl\{\mathbf{X}\in \operatorname{Gr}(l,n) : \mathbf{A}\subseteq \mathbf{X}\bigr\},\quad
\Omega_{-}(\mathbf{B})\coloneqq\bigl\{\mathbf{Y}\in \operatorname{Gr}(k,n) : \mathbf{Y} \subseteq \mathbf{B}\bigr\}.$$ We will call $\Omega_{+}(\mathbf{A})$ the *Schubert variety of $l$-planes containing $\mathbf{A}$* and $\Omega_{-}(\mathbf{B})$ the *Schubert variety of $k$-planes contained in $\mathbf{B}$*.

As we will see in Section \[sec:schubert\], $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ are indeed Schubert varieties and therefore closed subsets of $\operatorname{Gr}(l,n)$ and $\operatorname{Gr}(k,n)$ respectively. Furthermore, $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ are uniquely determined by $\mathbf{A}$ and $\mathbf{B}$ (see Proposition \[prop:Omega1\]) and may be regarded as ‘sub-Grassmannians’ of $\operatorname{Gr}(l,n)$ and $\operatorname{Gr}(k,n)$ respectively (see Proposition \[prop:OmegaGrass\]).

How could one define the distance between a subspace $\mathbf{A}$ of dimension $k$ and a subspace $\mathbf{B}$ of dimension $l$ in $\mathbb{R}^n$ when $k \ne l$? We may assume $k < l\le n$ without loss of generality. In which case a very natural solution is to define 