Abstract. A Čebyšev set in a metric space is one such that every point of the space has a unique nearest neighbour in the set. In Euclidean spaces, this property is equivalent to being closed, convex, and nonempty, but in other spaces classification of Čebyšev sets may be significantly more difficult. In particular, in hyperspaces over normed linear spaces several quite different classes of Čebyšev sets are known, with no unifying description. Some new families of Čebyšev sets in hyperspaces are exhibited, with dimension d + 1 (where d is the dimension of the underlying space). They are constructed as translational closures of appropriate nested arcs.1. Introduction. For any metric space (X, ρ), we define a set A ⊂ X to be a Čebyšev set ("be Čebyšev" or "have the Čebyšev property") if for every x ∈ X there is a unique nearest point in A. This property has been studied extensively for normed linear spaces. For such spaces, the Čebyšev property is related to convexity. For Minkowski spaces (finite-dimensional Banach spaces), every Čebyšev set is convex if balls are smooth, while if the balls are strictly convex, every nonempty closed convex set is Čebyšev [10]. In particular, in Euclidean spaces, the Čebyšev sets are precisely those that are nonempty, closed and convex.As a generalization of the Čebyšev property, we define A to be Čebyšev relative to X 0 if every point in X 0 has a unique nearest point in A [6]. This generalizes the concept of "reach": the reach of a set A can be defined