Let $\Omega$ be an open, bounded set in $\mathbb{R}^n$ with $C^1$ boundary. Is it true that the perimeter of the convex hull of $\Omega$ is smaller or equal the perimeter of $\Omega$ with equality if and only if $\Omega$ is a convex set? Can one always speak about the perimeter of the convex hull?

While it is probably true in the case $n=2$ it's false in dimension bigger than $2$. The counterexample I have in mind works as follows in $\mathbb{R}^3$. Take two disks parallel with the $xy$ plane and of the same radius centered on the $z$axis and fatten them up just a little bit. Built a thin cylindrical tube that connects them (along the $z$axis). The tube should be long enough, hence the disks should lie at quite a bit of distance. One can of course smooth the corners here without adding too much area. The convex hull will be a "cylinder" with radius equal to the radius of the original disks and height equal to the length of the tube plus two times the heights of the (fattened) disks. One can use appropriate values to check that the area of the convex hull is bigger than the area of the original set. 

