the question is the following: given a log-concave isotropic function $f$ defined on $\mathbb{R}^n$ and two compact 0-symmetric convex bodies $C_1, C_2$, where $C_1 \subset C_2$. Is it true that 
$$
  \frac{\int_{C_1} f}{Vol(C_1)} \geq  \frac{\int_{C_2} f}{Vol(C_2)}?
$$

We can also weaken it to 
$$
  \frac{Vol(C_1 \cap \mathcal{B}^n)}{Vol(C_1)} \geq \frac{Vol(C_2 \cap \mathcal{B}^n)}{Vol(C_2)},
$$

where $\mathcal{B}^n$ is the unit ball. Thank you.