Let $(C) \subset \mathbb{R}^3$ be a 3D solid cube with center $O = (0,0,0)$ and side length $=1$. Let $(H)$ be the projection of $(C)$ on the plane $z=0$, and $h$ be the smallest positive number satisfying $(C)$ is between two planes $z=h$ and $z=-h$ (points of $(C)$ are allowed to be on those planes).
I notice that, for several cases of $(C)$, the area of $(H)$ equals $2h$. So I propose a little conjecture that, indeed, $\text{Area}(H)=2h \; \; \forall (C)$.
I haven't proved this conjecture. Actually, I think that, if it's true, somebody definitely did notice and prove this fact. So I only post it here to learn more. Also, I want to propose some generalizations of this fact. Namely:
What are the necessary and sufficient conditions of $(C)$ so this fact still holds in $\mathbb{R}^3$?
How to rightfully generalize this fact into $n$-cubes in $\mathbb{R}^n$ ($n > 3$)?
What are the necessary and sufficient conditions of $(C)$ so this fact still holds in $\mathbb{R}^n$?
Thanks in advance.