Dido's Problem

Classic Problem
The new City of Carthage is to be founded.

Its area is to be determined by enclosing as much land as possible within a line whose length is fixed.

What shape should it be to make sure the area is a maximum?

Solution
The classical answer is that the enclosure was in the shape of a semicircle whose diameter is the coastline.

However, it is clear that this depends upon the shape of the coast, and a better solution may be to cut off a peninsula.