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.
If this is a very well-known fact from convex geometry, a reference to a book will be appreciated. Thank you.
