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?



An isoperimetric-type inequality inside a cube
This result is known as the *relative isoperimetric inequality*, see e.g. *Functions of Bounded Variation* by L. Ambrosio, N. Fusco and D. Pallara (2000), Eq. (3.43).


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 $\mbox{vol}(\Omega) \le \frac{1}{2}$.


