Imagine $N$ points $p_1,\cdots,p_N$ in the plane. Assume also that $$\|p_i-p_j\|\geq r\quad \forall j\neq i$$ for some $r>0$. Consider now a rotation of the axis i.e. $$\bar{p_i}=R\:p_i$$ for some rotation matrix $$R:=\left[\matrix{\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta)}\right]$$ I wonder if we can find an angle $\theta$ such that the minimum distance between the resulting $\bar{x}$ (or $\bar{y}$) coordinates be maximum. Specifically I am interested in $$\max_{\theta}\min_{\substack{i, j\\i\neq j}}|\bar{x}_i-\bar{x}_j|$$ with $\bar{x}_i$, $\bar{y}_i$ the coordinates of the $i$-th point in the new system i.e. $\bar{p}_i:=(\bar{x}_i,\bar{y}_i)$.
Edit: I am not necessarily looking for a sharp optimal value. If we can prove that there exists some $\theta$ such that $$\min_{\substack{i, j\\i\neq j}}|\bar{x}_i-\bar{x}_j|\geq \delta(N,r) r$$ for some $\delta(N,r)\in(0,1)$ this would be a nice step.
First let us consider one pair of points. We can rotate the coordinates so the $x$ axis is perpendicular to the line between them. The two points will then be $(x,a)$ and $(x,b)$ and the minimum coordinate distance is zero. As we rotate the coordinates they are at $(x\cos \theta - b \sin \theta, x \sin \theta+b \cos \theta)$ and $(x\cos \theta - a \sin \theta, x \sin \theta+a \cos \theta)$ with minimum distance $\min((a-b) \sin \theta, (a-b) \cos \theta)$ We will care about small angles, so the minimum distance will be $(a-b)\sin \theta$. You have said that $a-b \ge r$
You have $\frac 12N(N-1)$ pairs of points. Each pair will define four directions for the $+x$ axis where the minimum distance is zero. Intuitively, you want to rotate the coordinates so your axes are as far from these directions as possible. You can imagine plotting the $2N(N-1)$ directions for the $+x$ axis around the circle. There must be a gap of at least $\frac {\pi}{N(N-1)}$ radians between two neighboring points. The centerline of that gap will have a minimum difference of $r \sin \left(\frac {\pi}{2N(N-1)}\right)$. You can do somewhat better if the distance between the points on one side is greater than $r$ by moving $\theta$ in that direction or by choosing a smaller gap where the point spacings for the lines on each side is larger, but this is a lower bound.