Given a Poisson point process $$\mathcal{P}$$ in $$\mathbb{R}^2$$, the $$\textbf{Voronoi cells}$$ of a point $$p\in \mathcal{P}$$ is defined by $$V(x):=\{y\in \mathbb{R}^2: \|x-y\|=\min_{x'\in \mathcal{P}}\|x'-y\|\}$$ where $$\|\cdot\|$$ is the $$\ell^2$$ norm.
The Delaunay triangulation $$DT(\mathcal{P})$$ is the dual graph of the Voronoi diagram.(See figure as below) Given a square $$Q$$ with side length $$s(Q)=n$$ and $$\hat{Q}$$ with side length $$s(\hat{Q})=n/25$$ such that each cell of $$Q$$ that is divided into $$25^2$$ parts by cells of $$\hat{Q}$$. Assume that there exists at least one point in $$\hat{Q}$$ with high probability. If we define the $$\Gamma:=DT(\mathcal{P})\cap Q-A$$ where $$A\subset DT(\mathcal{P})\cap Q$$ and $$\partial_E(A_1, A_2)=\{e=\{x,y\}: x\in A_1, y\in A_2\}$$.
How to prove that isoperimetric inequality on Delaunay triangulation $$DT(\mathcal{P})$$ within a square $$Q$$, $$\mathbb{P}(\frac{|\partial_E(A, \Gamma)|}{|A|}\geq \frac{c}{n})\ge 1-e^{-cn}$$ for any $$A\subset DT(\mathcal{P})\cap Q$$ and $$|A|\le 0.5| DT(\mathcal{P})\cap Q|$$?