Deal with plane convex sets. A plane convex set, subsequently abbreviated to "convex s...

Deal with plane convex sets. A plane convex set, subsequently abbreviated to "convex set," is a nonempty set X in the plane having the property that if x and y are any two points in X, the straight-line segment from x to y is also in X. The following figures illustrate.

Prove that if X and Y are convex sets and X ∩ Y is nonempty, X ∩ Y is a convex set.