Consider a set of three non-orthogonal vectors $\vec{A}_{1}$, $\vec{A}_{2}$ and $\vec{A}_{3}$ such that norm of each of them is not equal to one i.e. $$||\vec{A}_{i}||\neq 1$$.
Consider a fourth vector $\vec{B}$ whose projection onto $\vec{A}_{i}$ is defined as $$P_{i}(B)=\vec{A}_{i}.\vec{B}$$ for $i=1,2,3$
Define $$P_{min}(\vec{B})=\min[P_{1}(B),P_{2}(B),P_{3}(B)]$$ The question is find $\vec{B}$ such that $P_{min}(\vec{B})$ takes the maximum value. i.e. find $\vec{B}^{*}$ such that $$P_{min}(\vec{B}^{*})=\max_{\vec{B}}P_{min}(\vec{B})$$
Intuitively, I think that for $\vec{B}^{*}$ , $P_{1}(\vec{B}^{*})=P_{2}(\vec{B}^{*})=P_{3}(\vec{B}^{*})$ but I cannot find any reason for it (apart from the fact that it is true when $\vec{A}_{i}$ are orthogonal).
Edit
I will give an argument for the support of my intuition below
First note that $\vec{B}^{*}$ will lie within the cone created by $\vec{A}_{1}$, $\vec{A}_{2}$ and $\vec{A}_{3}$. Because if the vector $\vec{B}$ do not lie within the solid angle made by $\vec{A}_{1}$, $\vec{A}_{2}$ and $\vec{A}_{3}$ then
$$P_{min}(\vec{B}) \leq 0$$
As $P_{min}(\vec{B}^{*})=\max_{\vec{B}}P_{min}(\vec{B}) > 0$ this implies that the vector $\vec{B}^{*}$ will only come from the cone made by $\vec{A}_{1}$, $\vec{A}_{2}$ and $\vec{A}_{3}$
Consider that for $\vec{B}^{*}$ which lies within the solid angle created by $\vec{A}_{1}$, $\vec{A}_{2}$ and $\vec{A}_{3}$ we have the following relation
$$\vec{A}_{1}.\vec{B}^{*} > \vec{A}_{2}.\vec{B}^{*} \geq \vec{A}_{3}.\vec{B}^{*}$$
By definition
$$P_{min}(\vec{B}^{*}) \geq P_{min}(\vec{B})$$
Now consider rotation of $\vec{B}^{*}$ with respect to the vector $\vec{A}_{2}$ in such a way that $$\vec{A}_{2}.\vec{B'}^{*}=\vec{A}_{2}.\vec{B}^{*}>\vec{A}_{1}.\vec{B'}^{*} \geq \vec{A}_{1}.\vec{B}^{*}$$
and
$$\vec{A}_{2}.\vec{B'}^{*} \leq \vec{A}_{3}.\vec{B'}^{*}$$
where $\vec{B'}^{*}$ is the vector obtained by the rotating $\vec{B}^{*}$ with respect to $\vec{A}_{2}$ axis.
Physically we can consider the rotated vector $\vec{B'}^{*}$ to be leaned slightly towards the $\vec{A}_{1}$ axis than $\vec{B}^{*}$.
Thus for the new vector we have $$P_{min}(\vec{B'}^{*}) \geq P_{min}(\vec{B}^{*})$$
violating the definition of $\vec{B}^{*}$. Thus, the condition $$\vec{A}_{1}.\vec{B}^{*} > \vec{A}_{2}.\vec{B}^{*} \geq \vec{A}_{3}.\vec{B}^{*}$$
cannot be valid. Which forces us to study whether $\vec{B}^{*}$ satisfies the condition
$$\vec{A}_{1}.\vec{B}^{*}=\vec{A}_{2}.\vec{B}^{*} > \vec{A}_{3}.\vec{B}^{*}$$
Proceeding in a similar fashion one can show that $\vec{B}^{*}$ cannot satisfy the above condition. Which implies that the only condition $\vec{B}^{*}$ will satisfy is
$$\vec{A}_{1}.\vec{B}^{*}=\vec{A}_{2}.\vec{B}^{*} = \vec{A}_{3}.\vec{B}^{*}$$
Comments and suggestions are welcome.