A bound for diameter of arithmetic hyperbolic orbifolds

Let $${\mathcal {O}}$$ be a closed n-dimensional arithmetic (real or complex) hyperbolic orbifold. We show that the diameter of $${\mathcal {O}}$$ is bounded above by $$\begin{aligned} \frac{c_1\log \mathrm {vol}({\mathcal {O}}) + c_2}{h({\mathcal {O}})}, \end{aligned}$$ where $$h({\mathcal {O}})$$ is the Cheeger constant of $${\mathcal {O}}$$ , $$\mathrm {vol}({\mathcal {O}})$$ is its volume, and constants $$c_1$$ , $$c_2$$ depend only on n.