Consider the following three points:
$$\mathbf{d}_1=\mathbf{v},~~~~~~~~ \\ \mathbf{d}_2=\mathbf{v}+\mathbf{e}_2,\\ \mathbf{d}_3=\mathbf{v}+\mathbf{e}_3,$$ where $\mathbf{v}$ is a fixed, but arbitrary 3D vector, while $\mathbf{e}_2=(0,1,0)^{\rm T}$ and $\mathbf{e}_3=(0,0,1)^{\rm T}$.
What is the global maximum of the following function:
$$f(\mathbf{u})=\min\limits_{~i=1 \\ \mathbf{u} \ne \mathbf{d}_i}^{3}\min\limits_{~j=1 \\ ~j \ne i}^{3}\frac{|(\mathbf{u}-\mathbf{d}_i)^{\rm T}(\mathbf{d}_j-\mathbf{d}_i)|}{\lvert\lvert\mathbf{u}-\mathbf{d}_i\rvert\rvert\,\lvert\lvert\mathbf{d}_j-\mathbf{d}_i\rvert\rvert},$$
where $\mathbf{u}$ is the variable vector with three enetries?
Update:
It seems to me that this problem can be divided into four subcases:
(1) $\mathbf{u} \notin \{\mathbf{d}_1,\mathbf{d}_2,\mathbf{d}_3\}$,
(2) $\mathbf{u}=\mathbf{d}_1$,
(3) $\mathbf{u}=\mathbf{d}_2$, or
(4) $\mathbf{u}=\mathbf{d}_3$.
Cases (2)--(4) seems to be trivial, so I would be mostly interested in case (1).