 $$\begin{aligned} \gamma(P_{n,k}) {\geqslant}\bigg(0- C\,\Big(c_0^{-0}+(\sqrt{\lambda_0({\mathcal{D}}_k)} + 0)\,h+ h^0\Big)\bigg) \,\bigg[\min_{i\in G_k^\ast}  \frac{d_{k,i}^\dagger}{d_{n,k}(X_i)}\bigg]\,\tau\,  C_k \, \lambda_0({\mathcal{D}}_k)\,h^0