Sierra Leone is the roundest country

Gonzalo Ciruelos set out to discover which country was the roundest in shape.

We can define roundness in many ways. For example, as you may know, the circle is the shape that given a fixed perimeter maximizes the area. This definition has many problems. One of the problems is that countries generally have chaotic perimeters (also known as borders), so they tend to be much longer than they seem to be.

For that reason, we have to define roundness some other way. We represent countries as a plane region, i.e., a compact set CaR2CaR2. I will define its roundness as

That's about where I tune out! Turns out the answer is Sierra Leone. Click through to see lots of mathy thingies on the screen, the runners-up, the least round countries, and the source code.