Let $\mathcal{X} = \{\mathbf{X}\in\mathbb{C}^{d\times d}:\|\mathbf{X}\|\leq 1\}$, where $\|\cdot\|$ is the Frobenius norm. Let $\mathbf{y}\in\mathbb{C}^{d\times 1}$. We are familiar with the following maximization: \begin{align} \max_{\mathbf{X}\in\mathcal{X}}\|\mathbf{X}\mathbf{y}\| \end{align} We have $\|\mathbf{X}\mathbf{y}\|\leq\|\mathbf{X}\|\|\mathbf{y}\| \leq \|\mathbf{y}\|$ with equality if, for example, $\mathbf{X} = \frac{\mathbf{y}\mathbf{y}^{\dagger}}{\|\mathbf{y}\|^2}$. Hence, we say that there is a rank-$1$ optimal solution. This is Cauchy-Scharwz in disguise and the result is quite intuitive. The way I see it is by taking the spectral decomposition $\mathbf{X}^{\dagger}\mathbf{X} = \sum_i \lambda_i \mathbf{v}_i\mathbf{v}_i^{\dagger}$. Then, $\|\mathbf{X}\mathbf{y}\|^2 = \sum_i \lambda_i |\langle \mathbf{y},\mathbf{v}_i\rangle|^2$. I have an orthonormal set of eigenvectors that I can optimize, with their respective "powers" (eigenvalues) that should sum up to at most $1$. The best way is to choose one of the $\mathbf{v}_i$ "in the direction of" $\mathbf{y}$ with the maximum power of $1$.

My question is concerned with the following generalization. Let $\mathbf{y}_1,\ldots,\mathbf{y}_K\in\mathbb{C}^{d\times 1}$ and consider \begin{align} \max_{\mathbf{X}\in\mathcal{X}}\min_{k\in\{1,\ldots,K\}}\|\mathbf{X}\mathbf{y}_k\| \end{align} The optimization problem discussed before is a particular case where $K=1$. Now, for any $K>1$, "intuitively," since we now have many vectors in many directions, we should always spend power in several different directions. Interestingly enough, when $K\in\{2,3\}$, there exists -again- a rank-$1$ optimal solution (see e.g. http://www1.se.cuhk.edu.hk/~zhang/Reports/seem2005-02.pdf). In general, there exists a roughly rank-$\sqrt{K}$ optimal solution, but the exact value appears to be open.

Similar to the case of $K=1$, I tried a lot to gain some geometric intuition on why rank-$1$ solutions would be optimal for $K=2$ or $K=3$, and why would the scaling be as at most $\sqrt{K}$ for large $K$, but failed to come up with a reasonable explanation. I was wondering if anyone has any ideas in this context.