A note on new weighted geometric inequalities for hypersurfaces in ${\mathbb{R}}^n$

In this note, we prove a family of sharp weighed inequality which involves weighted $k$-th mean curvature integral and two distinct quermassintegrals for closed hypersurfaces in ${\mathbb{R}}^n$. This inequality generalizes the corresponding result of Wei and Zhou \cite{WZ} where their proof is based on earlier results of Kwong-Miao \cite{KM1,KM2}. Here we present a proof which does not rely on Kwong-Miao’s results.