This is a very silly question.

For all regions S contained inside the unit square, what is the infimum of the quantity Perimeter(S)/Area(S)? This ratio being considered is not scale invariant, so it is only the constraint of being contained within the square which implies that this infimum is non-zero.

There are some "obvious" configurations to try, but I do not even know how to use a calculus of variations argument to show that these are local maxima.

Can you do better than $\displaystyle{2 + \sqrt{\pi}}$?