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Suppose $\mathbf{z}=(1,x,\mathbf{y}) \in \mathbf{R}^d$ where $x$ is a unknown scalar but $\underline{x}\le x\le\bar{x}$. Also, we know the exact value of vector $\mathbf{y}$ and $\|\mathbf{y}\|_2\le L_y$ and $\|\mathbf{z}\|_2\le L$.

If $\mathbf{V}$ is a PD matrix, what is the tightest possible upper bound for $\|\mathbf{z}\|_{\mathbf{V}^{-1}}$? I know that it is possible to find an upper bound like $L/\sqrt{\lambda_{\min}(\mathbf{V})}$, but it ignores all other information and other eigenvalues.

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