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In this paper, we prove weighted quantitative isoperimetric inequalities for the set    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$E_{\alpha}= \{(x,y)\in {R}^{h+1}: \vert y \vert <\int_{\arcsin \vert x \vert }^{\frac{\pi}{2}}\sin^{\alpha +1}(t)\,dt, \vert x \vert <1 \}$\end{document}    E  α   =  {  (  x  ,  y  )  ∈   R   h  +  1    :  |  y  |  <   ∫   arcsin  |  x  |    π  2sin   α  +  1    (  t  )    d  t  ,  |  x  |  <  1  }in half-cylinders in the Grushin space    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}${R}^{h+1}$\end{document}    R   h  +  1with density    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\vert x \vert ^{p}$\end{document}   |  x   |  p,    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$p\geq0$\end{document}   p  ≥  0.

Weighted quantitative isoperimetric inequalities in the Grushin space R h + 1 ${R}^{h+1}$ with density | x | p $|x|^{p}$