Impossibility of gathering by a set of autonomous mobile robots

Given a set of n autonomous mobile robots that can freely move on a two dimensional plane, they are required to gather in a position on the plane not fixed in advance (Gathering Problem). The main research question we address in this paper is: Under which conditions can this task be accomplished by the robots? The studied robots are quite simple: they are anonymous, totally asynchronous, they do not have any memory of past computations, they cannot explicitly communicate between each other. We show that this simple task cannot be in general accomplished by the considered system of robots.