12.11. (CSSR, 80). The set $M$ is obtained from the plane by removing three distinct points $A, B$, and $C$. Find the smallest number of convex sets whose union is the set $M$.
12.11. Let points $A, B$, and $C$ initially lie on the same line. Then these points divide the line into 4 intervals, and no points from different intervals can lie in the same convex set. Therefore, the number of required sets cannot be less than 4. The number 4 is achieved if the set $M$ is divided into parts as shown in Fig. 69.

![](https://cdn.mathpix.com/cropped/2024_05_21_03a501298610b72904feg-222.jpg?height=112&width=867&top_left_y=1010&top_left_x=77)

Fig. 69

Now, let points $A, B$, and $C$ not lie on the same line. Then points $B$ and $C$ divide the line $BC$ into 3 intervals, and points from different intervals must lie in different convex sets. Therefore, the number of required sets cannot be less than 3, and 3 sets can be chosen as shown in Fig. 70.