We will call a vertex $(\hat{\sigma},\hat{\tau})\in D$ to be of type $(k,l)$ if $\dim\sigma=k$ and $\dim\tau=l$. Let $\operatorname{star}_{\Sigma}(\hat{\sigma},\hat{\tau})$ be its star neighborhood in $\Sigma$. Let $\operatorname{vol}(\sigma)$ denote the normalized volume of the simplex $\sigma$ and same for $\operatorname{vol}(\tau)$.