Hausdorff geometry of polynomials

This is a survey of results and open problems in the geometry of polynomials. The basic idea is to find or to estimate the Hausdorff distance between two point sets on the complex plane corresponding to two polynomials.
For example, let $Z$ be the set of the zeros of a polynomial $p(z)$ and $Z'$ be the set of the zeros of $p'(z)$. By the Gauss–Lucas theorem, the Hausdorff distance $h(Z,Z')$ between $Z$ and $Z'$ is less or equal to the diameter $d$ of $Z$. Our conjecture is that $h(Z,Z')$ is less or equal to $d/2$.