Example 7 Given a line $l$ and two points $A$ and $B$ in the plane, how should a point $P$ be chosen on line $l$ so that $\max \{A P, B P\}$ is minimized?
Solution: We can assume that points $A$ and $B$ are on the same side of line $l$. If not, we can use the symmetric point $B'$ of point $B$ with respect to line $l$ for discussion.

Let the projections of points $A$ and $B$ on line $l$ be $A_1$ and $B_1$, respectively. Without loss of generality, assume $AA_1 \geqslant BB_1$.

If $AA_1 \geqslant BA_1$ (as shown in Figure 2), then $\max \{AA_1, BA_1\} = AA_1$. For any point $P$ on line $l$, we have $\max \{AP, BP\} \geqslant AA_1$. Therefore, point $A_1$ is the desired point.

If $AA_1 < BA_1$, then point $C$ lies on the segment $A_1B_1$ because $AA_1 \geqslant BB_1$. Clearly, $\max \{BC, AC\} = AC = BC$. For any point $P$ on line $l$, when $P$ is to the right of $C$, $\max \{AP, BP\} = AP > AC$; when $P$ is to the left of $C$, $\max \{AP, BP\} = BP > BC$. Therefore, point $C$ is the desired point.