The asymptotic consensus problem on convex metric spaces
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A consensus problem consists of a group of dynamic agents who seek to agree upon certain quantities of interest. The agents exchange information according to a communication network modeled as a directed time-varying graph. A convex metric space is a metric space on which we define a convex structure. Using this convex structure we define convex sets and in particular the convex hull of a (finite) set. In this paper we generalize the asymptotic consensus problem to convex metric spaces. Under minimal connectivity assumptions, we show that if at each iteration an agent updates its state by choosing a point from a particular subset of the convex hull of the agent's current state and the ones of his neighbors, the asymptotic agreement is achieved. As application example, we use this framework to introduce an iterative algorithm for reaching consensus of opinion. In this example, the agents take values in the space of discrete random variable on which we define an appropriate metric and convex structure. In addition, we provide a more detail analysis of the convex hull of a finite set for this particular convex metric space.