Let $\mathbf{\Phi}\in \mathbb{R}^{m\times n}$ be a real valued matrix with $m\le n$.
I want to find a vector $\mathbf{y}\in \mathbb{R}^m$ such that $\frac{\left|\langle \mathbf{y},\mathbf{\phi}_i\rangle\right|}{\|\mathbf{\phi}_i\|}=c$ for some constant $c$ for all $i=1,\cdots, n$, where $\mathbf{\phi}_i,\ i=1,\cdots, n$ are the columns of $\mathbf{\Phi}.$ However, I do not want $\frac{\langle \mathbf{y},\mathbf{\phi}_i\rangle\mathbf{\phi}_i} {\|\mathbf{\phi}_i\|^2}$ to be constant for all $i$.
Note that if $\mathbf{\phi}_i$ are all orthonormal, then such a vector is $\mathbf{y}=\mathbf{\Phi 1}\in \mathbb{R}^n$. However, I don't know where to start for the general case. Does anyone have any idea or maybe some pointers to relevant literature? I would appreciate any help. Thanks in advance.
Note: This is a special case of a general question asked here which also, sadly, remains unanswered.