Polar Convexity and Critical Points of Polynomials

A set A, in the extended complex plane, is called convex with respect to a pole u, if for any

the arc on the unique circle through x,y, and u, that connects x and y but does not contain u, is in A. If the pole u is taken at infinity, this notion reduces to the usual convexity. Polar convexity is connected with the classical Gauss-Lucas' and Laguerre's theorems for complex polynomials. If a set is convex with respect to u and contains the zeros of a polynomial, then it contains the zeros of its polar derivative with respect to u. A set may be convex with respect to more than one pole. The main goal of this article is to find the relationships between a set in the extended complex plane and its poles.