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If the norm of the difference between two unit vector is lower bounded by a positive constant, does it mean that the inner product is upper bounded?
Let $x,y$ be two vectors with $\lVert x \rVert = \lVert y \rVert =1$ and $\lVert x-y \rVert \geq \delta$, where $\delta \gt 0$. Is it possible to show that, $1-(x^Ty)^2 \geq \delta^2$?
My Approach:
$$\...
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alternative asymptotic bounds
I have an $n$ by 1 vector of weights $w$, and an $n$ by $k$ matrix, $\Gamma$. I have that $w'w$ is $\mathcal{O}(1)$, $\frac{\Gamma'\Gamma}{n}=\mathcal{O}(1)$ and $\frac{\Gamma\Gamma'}{n}=\mathcal{O}(1)...
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