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I know that $$ \min_{\|x\|_2=1=\|y\|_2} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq -1/2 $$ with equality whenever $|x_k|=\frac{1}{\sqrt{2}}=|y_k|$ for two coordinates.

I'm wondering if there's a sharper version that one can prove wherein $x$ is fixed and we only minimize over $y$, i.e., $$ \min_{\|y\|_2=1} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq\ ?? $$

Presumably the lower bound should depend on the sparsity of $x$, and something like its `variance' $$ \sigma(x) = \frac{1}{n}\sum_{k=1}^n(x_k-\mu(x))^2. $$

For example, in the initial problem, if $n$ is even and we take $|x_k|=\frac{1}{\sqrt{n}}$ with an alternating sign, then the minimum grows to $-1/n$. Here $\mu(x)=0$ and $\sigma(x)=1/n^2$. The minimum is achieved for $y$ with the same mean and variance. On the contrary, if $x_1=1$ (so $x$ has maximal variance) we find the minimum is 0 and is achieved for any $y$.

Are there any inequalities that reduce to these extreme cases but shed light on the intermediate ones as well?

I know that $$ \min_{\|x\|_2=1=\|y\|_2} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq -1/2 $$ with equality whenever $|x_k|=\frac{1}{\sqrt{2}}=|y_k|$ for two coordinates.

I'm wondering if there's a sharper version that one can prove wherein $x$ is fixed and we only minimize over $y$, i.e., $$ \min_{\|y\|_2=1} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq\ ?? $$

Presumably the lower bound should depend on the sparsity of $x$, and something like its `variance' $$ \sigma(x) = \frac{1}{n}\sum_{k=1}^n(x_k-\mu(x))^2. $$

For example, in the initial problem, if $n$ is even and we take $|x_k|=\frac{1}{\sqrt{n}}$ with an alternating sign, then the minimum grows to $-1/n$. Here $\mu(x)=0$ and $\sigma(x)=1/n^2$. The minimum is achieved for $y$ with the same mean and variance. On the contrary, if $x_1=1$ (so $x$ has maximal variance) we find the minimum is 0 and is achieved for any $y$.

Are there any inequalities that reduce to these extreme cases but shed light on the intermediate ones as well?

I know that $$ \min_{\|x\|_2=1=\|y\|_2} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq -1/n $$ with equality for even dimensions whenever $|x_k|=\frac{1}{\sqrt{n}}=|y_k|$.

I'm wondering if there's a sharper version that one can prove wherein $x$ is fixed and we only minimize over $y$, i.e., $$ \min_{\|y\|_2=1} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq\ ?? $$

Presumably the lower bound should depend on the sparsity of $x$, and something like its `variance' $$ \sigma(x) = \frac{1}{n}\sum_{k=1}^n(x_k-\mu(x))^2. $$

For example, in the initial problem the minimum was achieved for minimal variance $x$ and $y$. On the contrary, if $x_1=1$ (so $x$ has maximal variance) we find the minimum is 0 and is achieved for any $y$.

Are there any inequalities that reduce to these extreme cases but shed light on the intermediate ones as well?

I know that $$ \min_{\|x\|_2=1=\|y\|_2} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq -1/2 $$ with equality whenever $|x_k|=\frac{1}{\sqrt{2}}=|y_k|$ for two coordinates.

I'm wondering if there's a sharper version that one can prove wherein $x$ is fixed and we only minimize over $y$, i.e., $$ \min_{\|y\|_2=1} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq\ ?? $$

Presumably the lower bound should depend on the sparsity of $x$, and something like its `variance' $$ \sigma(x) = \frac{1}{n}\sum_{k=1}^n(x_k-\mu(x))^2. $$

For example, in the initial problem, if $n$ is even and we take $|x_k|=\frac{1}{\sqrt{n}}$ with an alternating sign, then the minimum grows to $-1/n$. Here $\mu(x)=0$ and $\sigma(x)=1/n^2$. The minimum is achieved for $y$ with the same mean and variance. On the contrary, if $x_1=1$ (so $x$ has maximal variance) we find the minimum is 0 and is achieved for any $y$.

Are there any inequalities that reduce to these extreme cases but shed light on the intermediate ones as well?