Gr\"unbaum's inequality for sections

We show \begin{align*} \frac{ \int_{E \cap \theta^+} f(x) dx }{ \int_E f(x) dx } \geq \left(\frac{k \gamma+1}{(n+1) \gamma+1}\right)^{\frac{k \gamma+1}{\gamma}} \end{align*} for all $k$-dimensional subspaces $E\subset {\mathbb{R}}^n$, $\theta \in E\cap S^{n-1}$, and all $\gamma$-concave functions $f:{\mathbb{R}}^n\to [0,\mathrm{\infty })$ with $\gamma >0$, $0< \int_{$\mathbb{R}$^n} f(x), dx <\infty$,and$\int_{$\mathbb{R}$^n} x f(x), dx$attheorigin$o\in$\mathbb{R}$^n$\mathrm{.}Here,$\theta^+ := \lbrace x, : , \langle x,\theta\rangle \geq 0 \rbrace$. As a consequence of this result, we get the following generalization of Gr"unbaum’s inequality: \begin{align*} \frac{ \mbox{vol}_k(K\cap E\cap\theta^+) }{ \mbox{vol}_k(K\cap E) } \geq \left( \frac{k}{n+1} \right)^k \end{align*} for all convex bodies $K\subset {\mathbb{R}}^n$ with centroid at the origin, $k$-dimensional subspaces $E\subset {\mathbb{R}}^n$, and $\theta \in E\cap S^{n-1}$. The lower bounds in both of our inequalities are the best possible, and we discuss the equality conditions.