It's not completely obvious why the projected perimeter of the inner box is bounded by the projected perimeter of the outer box, but it's a statement about one-dimensional segments that's easy enough to check.
What definition of "perimeter" are you using here?
Anyway if you define "perimeter" that way then for the outer box the "perimeter" is in fact 4 * (a+b+c), where a, b, c are the dimensions of the box. I also agree that for the inner box the "perimeter" is at least 4 * (a'+b'+c'), by the triangle inequality.
What is not at all clear to me is why the "perimeter" of the inner box is no larger than the "perimeter" of the outer box in this setup. You say it's easy to check, but it doesn't seem very obvious to me.
But this fact is, as you say, nontrivial. It's certainly false for various non-box-like shapes afaict, so there's something special to boxes that needs proving here.
I'm sorry, I see now that my original comment glossed over tons of stuff that was clear only to me. Instead of saying "easy to check" I should've posted my napkin sketch with the easy check.
Anyway, my solution is still simpler and more natural than the one in that pdf :-)