Given a set $S$ of non-zero vectors in $\mathbb{R}^n,$ and a subspace $L,$ consider $f(S,L)=\max_{s \in S}\frac{\|Ps\|}{\|s\|}$ where $Ps$ is the projection of $s$ onto $L.$

Specifically, consider the set $X$ consisting of the $2^n-1$ (nonzero) vectors with all coordinates $0$ or $1$.

A recent question concerns criteria which might show, for a given subspace $L$, that $f(X,L)$ is small. Here I am concerned with choosing the subspace:

For each $n$ and $d<n$, what is the minimum over all $d$-dimensional subspaces of $f(X,L)?$

If that seems too broad, then the case $d=1$ would be of interest.