1) Consider $n$ Points, $x_{1}, x_{2},...x_{n}$ distributed uniformly in $[0, 1]^d$. Term $d$ is the dimension.
2) Then, I construced a grid points $x^\prime_{1}, x^\prime_{2},...x^\prime_{n}$ that are the intersection of axes parallel lines with separation $n^{1/d}.$
Then, consider a ball or a sphere in the two cases with radius $r <1 $ at Points $x_{i}$ and $x^\prime_{i}$ .
I would like to know the number of Points $ N(x_{i})$ and $ N(x^\prime_{1})$ that fall in the ball with radius r for the two cases.
I found that in average we have $\mathbb{E} N(x_{i})=V nr^d$, where V is the volume of the d dimentional unit sphere.
And $\dfrac{1}{2d}\mathbb{E}N(x_{i}) \leq N(x^\prime_{i}) \leq \mathbb{E} N(x_{i})$.
However, I dont understand how this inequality can be true.