Gauss-Lucas Theorem states: "Let f be a polynomial and $f'$ the derivative of $f$. Then the theorem states that the $n-1$ roots of $f'$ all lie within the convex hull of the $n$ roots $\alpha_1,\ldots,\alpha_n$ of $f$."
My Question is: Is there a theorem which states that there exists a permutation $\sigma \in S_n$ that the inner area of the polygon which edges go through the roots of $f$ $\alpha_{\sigma(1)}\longrightarrow\alpha_{\sigma(2)}\longrightarrow\ldots\longrightarrow\alpha_{\sigma(n)}\longrightarrow\alpha_{\sigma(1)}$ contains all roots of $f'$?
EDIT (OB) It is not completely clear from the original question wether the OP allowed for self intersections of the polygonal curve with vertices the roots of $f$. The question that has a bounty on its head asks for a polygonal Jordan curve with vertices the roots of $f$ containing the roots of $f'$ further assuming $f$ has simple roots. Roots of $f'$ are allowed to lie on the edges of the polygonal Jordan curve. We further assume $n\geq 3$ and that the roots of $f$ are not all aligned (i.e. not all contained in a real affine line.)