Consider linear subspaces of $\mathbb{R}^n$. For two subspaces $X$ and $Y$, we define their Hausdorff distance as $$ {\displaystyle d_{\mathrm {H} }(X,Y)=\max \left\{\,\sup _{x\in X, |x|_2=1}\inf _{y\in Y,|y|_2=1}d(x,y),\,\sup _{y\in Y,|y|_2=1}\inf _{x\in X, |x|_2=1}d(x,y)\,\right\}.\!} $$
For $\epsilon>0$, the set of subspaces $X_1,\dots,X_m$ is called an $\epsilon$-net under Hausdorff distance if for any subspace $Y$, there is some $X_i$ such that $$ d_{\mathrm {H} }(X_i,Y)<\epsilon. $$ The questions is to provide upper and lower bound of $m$.
It is direct to observe that if $X$ and $Y$ have different dimensions, their distance is always greater than 1.
Therefore, we only need to consider $m=\sum_{k=1}^n m_k$, where $m_k$ denotes the $\epsilon$-net of fixed ($k$) dimensional subspaces.
Also, it would be interesting to consider another definition of Hausdorff distance as $$ {\displaystyle d_{\mathrm {H} }(X,Y)=\max \left\{\,\sup _{x\in X, |x|_2\leq 1}\inf _{y\in Y,|y|_2\leq 1}d(x,y),\,\sup _{y\in Y,|y|_2\leq1}\inf _{x\in X, |x|_2\leq 1}d(x,y)\,\right\}.\!} $$
Would the difference between the distance induces a significant difference between the bound of the $\epsilon$-net?
