# An isoperimetric-type inequality inside a cube

I am looking for a reference for the following inequality: if $$\Omega \subset [0,1]^d$$ satisfies $$\mbox{vol}(\Omega) \leq 1/2$$, then $$\mathcal{H}^{d-1}\left( \partial\Omega \cap (0,1)^d\right) \geq c_d \mbox{vol}(\Omega)^{\frac{d-1}{d}},$$ where $$\mathcal{H}^{d-1}$$ is the $$(d-1)-$$dimensional Hausdorff measure and $$c_d > 0$$ is a universal constant depending only on $$d$$.

This is a variation of the classical isoperimetric inequality with the interesting addition that surface 'on the boundary of the cube' does not count. This seems like it should be known. A discrete version of this inequality (for subsets of the grid graph) was proven by Bollobas and Leader (Edge-isoperimetric inequalities in the grid, Combinatorica 1991) and it seems there is a wealth of information for the discrete case. Has anybody seen the continuous case stated somewhere?

It follows from Poincare inequality (see e.g. Eq. (3.41) in the cited book) applied to $$\chi_\Omega$$ (the indicator of the set $$\Omega$$). Indeed, by Poincare inequality it holds $$\|\chi_\Omega - \mbox{vol}(\Omega)\|_{L^p((0,1)^d)} \le C |D\chi_\Omega|((0,1)^d)$$, where $$p=\frac{d}{d-1}$$. Here $$|D\chi_\Omega|((0,1)^d)=\mathcal{H}^{d-1}(\partial \Omega \cap (0,1^d))$$ if $$\partial \Omega$$ is sufficiently smooth. And $$\|\chi_\Omega - \mbox{vol}(\Omega)\|_p = \bigl((1 - \mbox{vol}(\Omega))^p \mbox{vol}(\Omega) + \mbox{vol}(\Omega)^p (1 - \mbox{vol}(\Omega))\bigr)^{1/p} \ge \frac{1}{2} \mbox{vol}(\Omega)^{1/p}$$ since $$\mbox{vol}(\Omega) \le \frac{1}{2}$$.