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Convex Bodies of Constant Width and the Apollonian Metric
Marina Borovikova and Zair Ibragimov
Abstract.
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.