Convex Bodies of Constant Width and the Apollonian Metric

The study of constant width sets goes at least as far back as the time of Euler. The Apollonian metric, on the other hand, is a relatively new concept. It was introduced by Beardon in 1998 as a generalization of the hyperbolic metric of a ball to arbitrary domains [3]. Close connections between these concepts were established in [20] and [21]. In this paper, we study the Apollonian metric of domains which are the complements of constant width sets. We verify Beardon’s conjecture for such domains and show that in such domains the circular arcs which are orthogonal to the boundary and only they are the pseudogeodesic lines.