Intersection of 4 convex sets in $\mathbb{R}^2$

Question

Does there exist an example of four convex sets lying in $\mathbb{R}^2$ such that the intersection of every 3 of them contains a unit-length interval but the intersection of all 4 of them does not?

Answer 1

Yes. Consider the four rectangles

(0 , 0), (1 , 0), (1 ,1.5), (0 ,1.5) (0.5, 0), (1.5, 0), (1.5,1.5), (0.5,1.5) (0 , 0), (0 , 1), (1.5, 1), (1.5, 0) (0 ,0.5), (0 ,1.5), (1.5,1.5), (1.5,0.5)

Then the intersection of each triple is a $1 \times 0.5$ (or $0.5 \times 1$) rectangle, and the intersection of all four is a $0.5 \times 0.5$ square.