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I need help with the derivation of the inequality on the third line from bottom to top in a proof of Borsuk's problem, there are no explanation for that step. Thanks in advance.

http://i.stack.imgur.com/QopXS.png

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Do you mean the $m\leq n+1$? That's probably a typo (and I'd presume they meant $d$ instead of $n$). –  anonymous Oct 14 '12 at 19:17

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Well, as the text says, $S_\rho$ is a sphere of radius $\rho$ and centered at the origin, so if $a_i\in S_\rho$, then

$$\langle a_i,a_i\rangle=\text{square of distance from origin}=\text{square of radius of the sphere}.$$

Moreover, $\langle a_i,a_j\rangle\geq\min\langle a_i,a_j\rangle=-\varepsilon\rho^2$.

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Explain the first part comprehensively please, i also don't know the definition of S_p . –  Victor Oct 14 '12 at 19:21
    
@Victor: I added some more explanation –  enzotib Oct 14 '12 at 19:26

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