We consider the following inequality: \t\t\tμ(L)n−kn≤CkmaxH∈Grn−kμ(L∩H),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\mu (L)^{\\frac{n-k}{n}} \\leq C^{k}\\max_{H\\in \\mathit{Gr}_{n-k}}\\mu (L \\cap H), \\end{aligned}$$ \\end{document} which is a variant of the notable slicing inequality in convex geometry, where L is an origin-symmetric star body in {{mathbb{R}}}^{n} and is μ-measurable, μ is a nonnegative measure on {mathbb{R}} ^{n}, mathit{Gr}_{n-k} is the Grassmanian of an n-k-dimensional subspaces of {mathbb{R}}^{n}, and C is a constant. By constructing the generalized k-intersection body with respect to μ, we get some results on this inequality.