Deforming star-shaped polygons in the plane

Given a star-shaped polygon in the euclidean plane and a vertex v of the polygon, the author characterizes all those points w in the plane such that when the vertex v moves to w along a straight line path, while all other vertices of the polygon are fixed, the polygon remains star-shaped. An example is given to show that this characterization fails for the star-shaped polyhedral spheres in the 3-dimensional euclidean space.