On the apollonian metric of domains in

We study the apollonian metric considered for sets in ? _n by Beardon in 1995. This metric was first introduced for plane Jordan domains by Barbilian in 1934. For a special class of plane domains Beardon showed that conformal apollonian isometries are Möbius transformations. We give here a proof of Beardon's result without conformality assumption. We show that the apollonian metric of a domain D is either conformal at every point of D, at only one point of D or at no point of D. We also present a suprising relation between convex bodies of constant width and the apollonian metric.