What are some examples of isotrophic sets?

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>^2 )= ||x||_2 ^2$

Answer 1

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...

Answer 2

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, $y=\sqrt{abs(x)-x^2}$