Does there exist a compact subset $K$ of $\mathbb{R}^2$ with the following property? 

Let $x$ and $y$ be a pair of points in $K$ such that \begin{equation} |x-y| < \sup_{z,w \in K}|z-w| \end{equation} Then it is possible to rotate $K$ around its center to obtain a set $K'$ such that $K'$ still contains both $x$ and $y$ and such that at least one if $x$ and $y$ lies in the interior of $K'$. You can take pretty much any definition you want for the 'center' of $K$.