One thing I've kind of seen of all over the shop is that in $\mathbb{R}^{2}$ taking the convex hull of a bounded open set whose closure is connected reduces perimeter and increases area.
I can see the argument in the cases of polygons and for sets that are closure of the interior of some simple closed differentiable curve from an intuitive argument. However, it is not clear to me how one could show this in generality. I feel like I may just be missing some startlingly obvious point.
Understandably I should mention that I'm using the 1-dimensional Hausdorff measure of the topological boundary for perimeter, but of course if someone has an elegant way using De Giorgi perimeter then that would also hold.
Many thanks,
Cryptokyo