A general slicing inequality for measures of convex bodies

Abstract

Abstract We consider the following inequality: μ(L)n−kn≤CkmaxH∈Grn−kμ(L∩H), $$\begin{aligned} \mu (L)^{\frac{n-k}{n}} \leq C^{k}\max_{H\in \mathit{Gr}_{n-k}}\mu (L \cap H), \end{aligned}$$ which is a variant of the notable slicing inequality in convex geometry, where L is an origin-symmetric star body in Rn ${{\mathbb{R}}}^{n}$ and is μ-measurable, μ is a nonnegative measure on Rn ${\mathbb{R}} ^{n}$, Grn−k $\mathit{Gr}_{n-k}$ is the Grassmanian of an n−k $n-k$-dimensional subspaces of Rn ${\mathbb{R}}^{n}$, and C is a constant. By constructing the generalized k-intersection body with respect to μ, we get some results on this inequality.

Keywords

• Convex bodies
• Intersection bodies
• Generalized measures