For a subset $S$ of $\mathbb{R}^n$, we denote by $\lambda S$ the dilation for any $\lambda \in \mathbb{R}$: $$\lambda S=\{\lambda x| x\in S\}.$$
Let $\Omega$ be a convex body in $\mathbb{R}^n$ with $0$ as barycenter. Put $$a(t)=\frac{vol(t\Omega \bigcap -(2-t)\Omega)}{vol(\Omega)}(0\leq t\leq 1).$$ Then we want to know whether the following holds: $$\int_0^1 a(t)dt \geq \frac{n!2^n}{(n+1)^{n+1}}.$$
I don't know the answer even for $n=2$. When $\Omega$ is a triangle,
\begin{equation} a(t) = \left\{ \begin{array}{ll} t^2 & 0\leq t \leq \frac{2}{3} \\ -2t^2+4t-\frac{4}{3} & \frac{2}{3} \leq t\leq 1 \end{array} \right. \end{equation} So $$\int_0^1a(t)dt=\frac{8}{27}.$$ It's known that $a(1)\geq \frac{2}{3}$ for convex polygon when $n=2$.