Rogers-Shepard Type Inequalities for Sections

In this paper we address the following question: given a measure $\mu$ on ${\mathbb{R}}^n$, does there exists a constant $C>0$ such that, for any $m$-dimensional subspace $H\subset {\mathbb{R}}^n$ and any convex body $K \subset $\mathbb{R}$^n$, the following sectional Rogers-Shephard type inequality holds: $$\mu ((K-K)\cap H)\le C\underset{y\in {\mathbb{R}}^n}{\sup }\mu (K\cap (H+y))?$$ We show that this inequality is affirmative in the class of measures with radially decreasing densities with the constant $C(n,m)=\left(\genfrac{}{}{0px}{}{n+m}{m}\right)$. We also prove marginal inequalities of the Rogers-Shephard type for $\left(\frac{1}{s}\right)$-concave, $0\le s<\mathrm{\infty }$, and logarithmically concave functions.