I have a convex polytope $P$ in $R^n$. I cut it with $n$ orthogonal hyperplanes, each one passing through the center of gravity,, thus obtaining $2^n$ polytopes. W.r.t. the measure (volume) of $P$, what is the measure of the largest cut?
In $n=2$, it is easy to see that the largest cut can almost attain half of the area (instead of a quarter for a perfect cut). What about higher dimensions?
Making an answer out of Tal-Botvinnik's comment.
In $\mathbb{R}^3$, consider an elongated cylinder of radius $r > 0$ and height $h$, with $h$ much bigger than $r$. If $d = (1,1,1)$ is the direction of the height and $V$ the volume of the cylinder intersected with the corresponding quadrant, you can find the simple bound $$ V \geq \pi r^2 (h/2 - \sqrt{3}r). $$ For a fixed volume of the cylinder, as $r\rightarrow 0$, $V$ tends to half the volume of the cylinder.
The same argument works in $\mathbb{R}^n$, replacing $\pi r^2$ by the area of a disk of radius $r$ and $\sqrt{3}$ by $\sqrt{n}$.