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: $$\mu((K-K) \cap H) \leq C \sup_{y \in \mathbb{R}^n} \mu(K \cap (H+y))?$$ We show that this inequality is affirmative in the class of measures with radially decreasing densities with the constant $C(n,m) = \binom{n+m}{m}$. We also prove marginal inequalities of the Rogers-Shephard type for $\left(\frac{1}{s}\right)$-concave, $0 \leq s < \infty$, and logarithmically concave functions.