Exercise: Find two Borel disjoint and bounded sets $E, F \subset \mathbb R^n$ such that $\operatorname{Per}(E) + \operatorname{Per}(F) > \operatorname{Per}(E \cup F)$.
($\operatorname{Per}(A)$ is the perimeter of the set $A$).
Attempt: I can solve this without the request "disjoint", it's pretty easy, but not in the general case! I would pick two open squares of lenght 1 sharing a face, but I don't even know if the perimeter of an open set makes sense.
In one dimension, $(0,1)\cup [1,2) = [0,2)$ is such an example: the perimeter of every interval is equal to $2$.
In higher dimensions $n>1$, multiply the above example by a cube $(0,1)^{n-1}$. The perimeter of a rectangular box is the same regardless of the box being open, closed, or half-closed: indeed, the Caccioppoli definition of perimeter shows it is not affected by adding or removing a set of zero $n$-dimensional measure.