PomeLemma 19.7.10.09E448752D43 Assume that $n>0$. The maps $\mathcal{K}_ n\rightarrow \mathbb {R}$ given by
\[ K\mapsto \min _{x\in K}x_1,\qquad K\mapsto \max _{x\in K}x_1 \]
are continuous with respect to the Hausdorff metric.
Lemma 19.7.1009E448752D43
Proof. We prove the assertion for the maximum; the proof for the minimum is the same. Let $K,L\in \mathcal{K}_ n$ and set $\epsilon =d_{\mathrm{Haus}}(K,L)$. For every $x\in K$, there exists $y\in L$ with $|x-y|\leq \epsilon $, hence
\[ x_1\leq y_1+\epsilon \leq \max _{z\in L}z_1+\epsilon . \]
Taking the maximum over $x\in K$ gives
\[ \max _{x\in K}x_1\leq \max _{z\in L}z_1+\epsilon . \]
Interchanging $K$ and $L$, we get
\[ \left|\max _{x\in K}x_1-\max _{z\in L}z_1\right|\leq d_{\mathrm{Haus}}(K,L). \]
This proves the continuity. $\square$
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