Originally Posted by

**Runty** Our textbook has no knowledge about a hyper-surface (or the word is mixed into a random paragraph which I will never find), so I could use some help with this one. I have a potential answer for the second half of this question, which boils down to the AM-GM Inequality, but I'm not sure if my answer is what is being asked for (the context of the question is a bit vague to me).

Find the minimum distance from the origin to a point on the hyper-surface

$\displaystyle x_1x_2...x_n=p^n$, $\displaystyle p>0$

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I've been told by another source that, since a hyper-surface is a surface of $\displaystyle n$ dimensions, the minimum distance would occur along the line $\displaystyle x_1=x_2=...=x_n$, which would mean when they all equal $\displaystyle p$. However, showing this is proving difficult for me.

Deduce that, for $\displaystyle a_i\geq 0$,

$\displaystyle \frac{1}{n}\sum_{i=1}^n a_i\geq \sqrt[n]{a_1a_2...a_n}$

with equality occurring only when all of the $\displaystyle a$'s are equal.

(HINT: look at $\displaystyle a_i=x^2_i$)

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For this second half, this is basically the inequality of arithmetic and geometric means (or the **AM-GM inequality**), for which I could easily find a proof. However, due to how the question is worded, I'm not entirely sure if a proof is what is being asked of us. As such, I could use some clarification.

Any help on this one would be appreciated.