MathNet 2010 — Part 07 Question 31

Let $S$ be a convex set in the plane with a finite area $a$. Prove that either $a=0$ or $S$ is bounded.

Note: a set is bounded if it is contained in a circle of finite radius.

Note: a set is convex if, whenever two points $A$ and $B$ are in the set, the line segment between them is also in the set.