I have the following geometrical problem:
OBSERVATION Consider the unit cube in 3 dimensions, and N orthogonal parallelepiped of size 1/N x 1/N x 1 (hence each of volume $V=1/N^2$). It is easy to place these N orthogonal parallelepiped inside the cube such that if one takes two points of coordinates (x,y,z) and (x',y',z') in two distinct parallelepiped, it holds that $x\neq x', y\neq y', z\neq z'$. For instance, the orthogonal parallelepiped number j is composed of the points (x,y,z) such that $j/N<x,y<(j+1)/N$ and $0<z<1$.
QUESTION Assume that the unit cube in 3 dimensions contains N objects of same volume V such that if one takes two points of coordinates (x,y,z) and (x',y',z') in two distinct objects, it holds that $x\neq x', y\neq y', z\neq z'$. Then:
- a. Can I show that $V\leq 1/N^2$?
- b. If yes and if $V=1/N^2$, must the objects be orthogonal parallelepiped of size 1/N x 1/N x 1. How can they be placed?
- c. is there an inequality with an associated equality condition capturing all this?