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What are some examples of isotrophic sets? and is there a "good" way to describe them?

Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all $x \in R^n $ expected value $ E( < x,X>^2 )= ||x||_2 ^2$

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  • $\begingroup$ Do you mean to say $E(|\langle x, X\rangle|)=\|x\|$? $\endgroup$ – Eric Wofsey Dec 30 '14 at 8:09
  • $\begingroup$ Yes I mean $ E(|<x,X>|)= ||x||_2 $ $\endgroup$ – Hao S Dec 30 '14 at 19:40
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    $\begingroup$ Eric Wofsey asked if you meant to refer to the absolute value of the inner product, but you changed your question instead to ask about the square of the inner product. Which do you mean? $\endgroup$ – LSpice Dec 30 '14 at 21:49
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This answer is referring to version 1 of the question.

What are some examples of isotrophic sets? and is there a "good" way to describe them?

Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all $x \in R^n $ expected value $ E( < x,X> )= ||x||_2$

Let $i=(1,0)$ and $j=(0,1)$. Then $$ 2 = 1 + 1 = \|i|_2 + \|j\|_2 = E(i\cdot X) + E(j\cdot X) = E(i\cdot X + j\cdot X) = E((i+j)\cdot X) = \|i+j\|_2 = \sqrt{2}$$ so there cannot be such an $X$. So I guess the set does not have any finite area...

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Examples of such sets $X$ can be generated knowing that $\|\lt u,v\gt\|_2 = \|u\|_2\|v\|_2\cos(u,v)$ and thus that if $\|v\|_2 = \frac{1}{\cos(u,v)}$, we have $\|\lt u,v\gt\|_2=\|u\|_2$.

if $u\in\mathbb{R}^1$ and $v=\left(\sqrt{s^2+t^2}\frac{1}{\sqrt{1+\frac{t}{s}}},\sqrt{s^2+t^2}\frac{\frac{t}{s}}{\sqrt{1+\frac{t}{s}}}\right)\in\mathbb{R}^2,s,t\in\mathbb{R}$, then $X:=\{v\}$ is a concrete example.

Edit:
for $u\in\mathbb{R}^1$ and $v\in\mathbb{R}^2$ we have
$\|v\|_2=\sqrt{x^2+y^2}$ and $cos(0,v)=\cos(\arctan(\frac{y}{x}))=\frac{abs(x)}{\sqrt{x^2+y^2}}$;
from $\|v\|_2=\frac{1}{\cos(0,v)}$ we get $\sqrt{x^2+y^2}=\frac{abs(x)}{\sqrt{x^2+y^2}}$ and finally, $y=\sqrt{abs(x)-x^2}$

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  • $\begingroup$ The question asks for this to hold for all $x$, not just a single $x$. $\endgroup$ – Eric Wofsey Dec 30 '14 at 9:10

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