An isoperimetric-type inequality inside a cube

Question

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?

Answer 1

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