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 $\displaystyle{2 + \sqrt{\pi}}$?

EDIT: One can probably show that an optimal S has $\mathbf{Z}/4\mathbf{Z}$ symmetry. Assume that S consists of line segments from [x,1-x] on the boundary, and then (four quarters of) a shape T in the corner. We can assume that T has volume Ax^2 and perimeter Px. Minimizing the quantity with respect to x, one obtains a minimum of:

$$2 +\sqrt{4 + \frac{(P/2-4)^2}{(A - 4)}},$$

which becomes $2 + \sqrt{\pi}$ when T is the circle, i.e., $A = \pi$ and $P = 2 \pi$. Attempts to improve upon a circle for T have failed.