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Let $S$ be a finite subset of $R^n$. Let $\widehat{x}$ indicate the mean (non-zero) of the points in $S$. Is there an $S$ such that $\widehat{x}$ is orthogonal to all the points in $S$?

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  • $\begingroup$ What does it mean for points to be orthogonal to each other? Do you mean the corresponding vectors? $\endgroup$
    – joriki
    Nov 10, 2012 at 12:23
  • $\begingroup$ @joriki, yes the corresponding vectors. $\endgroup$ Nov 10, 2012 at 12:26

2 Answers 2

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No, there isn't. If $\hat x$ is orthogonal to all vectors in $S$, then it's also orthogonal to itself, and thus zero.

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  • $\begingroup$ I upvoted your answer but accepted martini's answer for its detailed description :-) I hope that is fine. $\endgroup$ Nov 10, 2012 at 12:51
  • $\begingroup$ @Tenali: Sure, whatever was more useful to you :-) $\endgroup$
    – joriki
    Nov 10, 2012 at 12:51
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Let $S \subseteq \mathbb R^n$. We denote by $U := \operatorname{span} S$ the linear hull of $S$. If $y \mathbin \bot S$, then by linearity of the scalar product, $y \mathbin \bot U$. No, if on one hand $\hat x = \frac 1{|S|}\sum_{x\in S} x \in U$, on the other $\hat x \mathbin\bot S$, we have $\hat x \mathbin \bot U$, so $\hat x \mathbin \bot \hat x$. That is $\hat x = 0$.

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