Take a set of points $X = \{x_1,\ldots,x_n\} \subset [0,1]^d$ that includes the vertices of the unit hypercube $\{0,1\}^d$. Is there a known lower bound for $$ \max_{x\in X} \min_{y \in X} \|x-y\|_2\;, $$ where $\|\cdot\|_2$ is the usual Euclidean norm?
In words, I am wondering if given $n$ points in the unit hypercube (I am including the vertices to avoid the trivial case in which all points are arbitrarily closed to each other) one can guarantee that the distance between (at least a pair of) two closest points is larger than something.
Intuitively, I think it should be of order $1/n^{1/d}$ because I imagine the hardest case is when the points form a lattice. However, I have no idea how to prove it formally.