Fix a convex $n$-gon $\Gamma$, is it true that for any subset $E$ of $\mathbb{R}^2$ with positive Lebesgue measure, there exist $n$ points in $E$, such that the associated $n$-gon of these points is similar to $\Gamma$ ? If this is true, could this be generalized to $\mathbb{R}^m$ with $m >= 3$?
I have already proved that the answer is affirmative when $\Gamma$ is regular, by considering rotations of $\mathbb{R}^2$ by $\frac{n}{2 \pi}$, and applying a lemma which asserts the existence of a square $S_{\epsilon}$ satisfying $\frac{m(S_{\epsilon} \cap E)}{m(S_{\epsilon})}>1-\epsilon$ for $\epsilon > 0$.
Any remark will be greatly appreciated.
I see, direct application of (almost) the same method yields the desired result, namely for any ${p_{1},...,p_{n}}$ with prescribed similar type we consider still a order-$n$ subgroup of $Aut_{Top}(\mathbb{R^2})$ fixing ${p_{1},...,p_{n}}$, and let it act on a region containing ${p_{1},...,p_{n}}$.