Example 4 (Beijing) Let $D$ be the set of points consisting of the equilateral $\triangle P_{1} P_{2} P_{3}$ and its interior, and let point $P_{0}$ be the center of $\triangle P_{1} P_{2} P_{3}$. If the set $S=$ $\left\{P|P \in D|, P P_{0}|\leqslant| P P_{i} \mid, i=1,2,3\right\}$, then the planar region represented by the set $S$ is ( )
A. Triangular region
B. Quadrilateral region
C. Pentagonal region
D. Hexagonal region
As shown in the figure, $A, B, C, D, E, F$ are the trisection points of each side. From $\left|P P_{0}\right| \leqslant\left|P P_{1}\right|$, we know that point $P$ is on the perpendicular bisector of segment $P_{0} P_{i}$ or on the side closer to point $P_{0}$. Since point $P \in D$, then $S$ represents the regular hexagon $A B C D E F$ as shown in Figure 2. Choose D.