Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ of dimension $m$.
We fix an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak g$.  

We may assume (in case is necessary) that $G$ is semisimple and/or $\langle\cdot,\cdot\rangle$ is $\textrm{Ad}(G)$-invariant. 


Given any subspace $\mathcal H$ of $\mathfrak g$, $(\mathcal H,\langle\cdot,\cdot\rangle|\_{\mathcal H})$ induces a left-invariant sub-Riemannian structure on $G$. 
The corresponding sub-Riemannian manifold $(G,\mathcal H,\langle\cdot,\cdot\rangle|\_{\mathcal H})$ has associated a distance $d\_{\mathcal H}$ on $G$ known as the Carnot–Carathéodory metric. 
For $a,b\in G$, $$d\_{\mathcal H}(a,b)=\inf\_{\gamma} \int\_0^1 \langle \gamma'(t),\gamma'(t)\rangle^{1/2}\, dt,$$ where the infimum is taken over all smooth curves $\gamma:[0,1]\to G$ such that $\gamma(0)=a$, $\gamma(1)=b$, and $\gamma'(t)\in \mathcal H$ for all $t$. 
Chow's theorem ensures that $d\_{\mathcal H}(a,b)<\infty$ for all $a,b\in G$ if $\mathcal H$ is bracket-generating (i.e. the only subalgebra of $\mathfrak g$ containing $\mathcal H$ is $\mathfrak g$). 
I am interested in the diameter of the metric space $(G,d\_{\mathcal H})$, i.e. $$\textrm{diam}(G,d\_{\mathcal H})=\max\_{a,b\in G}d\_{\mathcal H}(a,b).$$


Let $\textrm{Gr}\_{\mathfrak g}(k)$ denote the set of $k$-dimensional subspaces of $\mathfrak g$. 
This is a symmetric space called *Grassmannian*. 
We will only use the topology on it induced by the distance metric associated with the Riemannian symmetric metric. 



> 
> Is the map $\Phi: \textrm{Gr}\_{\mathfrak g}(k)\to \mathbb R\_{\geq 0}\cup\{\infty\}$ given by $\Phi(\mathcal H)= \textrm{diam}(G,d\_{\mathcal H})$ continuous?
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> 
> 


A reference or a proof would be very appreciated. 



On the diameter of left-invariant sub-Riemannian structures on a compact Lie group
From semicontinuity of length in $(G,\langle\ ,\ \rangle)$, we get that diameter is lower semicontinuous.
It remains to show that it is upper semicontinuous.


Suppose $\mathcal H\_n\to \mathcal H\_\infty$ as $n\to\infty$.
We can assume that $H\_\infty$ is bracket-generating, otherwise $(G,\mathcal H\_\infty,\langle\ ,\ \rangle)$ has infinite diameter --- so, there is nothing to prove.


Suppose two points $x$ and $y$ are connected by a path $\gamma$ of length $\ell$. We can assume that $\gamma\_\infty$ is Lipschitz, in particular it is differentiable almost everywhere and it completely defined by its derivative $\gamma\_\infty'\colon t\to H\_\infty$.
Denote by $\pi\_n$ the orthogonal projection $\mathfrak{g}\to \mathcal H\_n$.
Consider the curve $\gamma\_n$ that starts at $x$ and $\gamma'\_n(t)=\pi\_n\circ\gamma\_\infty'(t)$ for any $t$.
Observe that
$$\mathrm{lenght}\,\gamma\_n\leqslant \mathrm{lenght}\,\gamma\_\infty$$
for any $n$.
Further note that $\gamma\_n\to \gamma\_\infty$ as $n\to\infty$;
in particular $y\_n=\gamma\_n(1)\to y$ as $n\to\infty$.


Since $H\_\infty$ is bracket-generating, so is every $H\_n$ for every large $n$.
Moreover, for any $\varepsilon>0$ we can choose a neighborhood $N$ of $y$ such that for any large $n$, the distance in $(G,\mathcal H\_n,\langle\ ,\ \rangle)$ from $y$ to any $z\in N$ is smaller than $\varepsilon$.
It follows that given $\varepsilon>0$, the distance from $x$ to $y$ in $(G,\mathcal H\_n,\langle\ ,\ \rangle)$ exceeds the distance in $(G,\mathcal H\_\infty,\langle\ ,\ \rangle)$ is at most $\ell+\varepsilon$.
Hence the upper semicontinuity follows.


