A finite exact algorithm for epsilon-core membership in two dimensions

Let V be a finite configuration of voter ideal points in the Euclidean plane. For given > 0 a point x ∈ < is in the -core if for all y 6= x, ||v−x|| ≤ ||v− y||+ for a simple majority (at least |V |/2) of voters v ∈ V . Let (x) denote the the least for which x is in the -core. Thus (x) = 0 if and only if x is a core point. The least for which the -core is nonempty is denoted ∗. This paper provides a finite algorithm, given V , x, and , to determine whether x is in the -core. By bisection search, this yields a convergent algorithm, given V and x, to compute (x). If the function (x) were strictly convex this would lead to a convergent algorithm to compute ∗ and the corresponding point. However, we prove that the function (x) is not convex in general.