This is a problem in integral geometry. I can't prove or find a proof of the following inequality :
Let $\beta$ be a simple closed curve and $Box(r)=\{|x|< r, |y|< 2^{1/3}r, |z|< 2^{2/3}r\}\subseteq \mathbb{R}³$ and $\epsilon>0$ be fixed. Then $$ \int_1^{1+\epsilon} \#(\beta \cap \partial Box(r))dr \leq Length(\beta \cap Box(1+\epsilon)) $$
It sounds like Crofton's formula and isn't hard to believe in $\mathbb{R}²$ with circles instead of boxes. Does anyone have a reference or a proof?
Thanks a lot.