On the apollonian metric of domains in

Abstract

We study the apollonian metric considered for sets in R 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 Mo¨bius 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.

Barbilian’s metrization procedure in the plane yields either Riemannian or Lagrange generalized metrics

“…In the view of Theorem 2, it might be of interest to note that Zair Ibragimov has shown that the Barbilian metric is Riemannian in zero, one or all points of the domain (he uses the term "conformal" for Riemannian). See [24], [25] for this fact and related implications.…”

“…Later contributions on the topic of Barbilian spaces include P. J. Kelly's work [26] and major developments are due to D. Barbilian himself [3], [4], [5], [6]. Recently, many citations of the work [2], originally published in Časopis Mathematiky a Fysiky, have appeared, for example, in [7], [11], [13], [16], [18], [19], [20], [21], [22], [23], [24], [25]. The history of this subject is presented in [14], [15].…”

“…Over the years, the paper [2] has been cited many times (e.g. [7], [8], [9], [10], [12], [13], [18], [19], [20], [21], [22], [23], [24], [25]). The geometric viewpoint is discussed in the monograph [11].…”

Barbilian spaces: The history of a geometric idea

“…This note is motivated by the recent increased attention to Barbilian's work (see the references at the end to Beardon, 1998;Boskoff, 1993Boskoff, , 1994Boskoff, , 1995Boskoff, , 1996aBoskoff, , 1996bBoskoff, , 1998Boskoff, , 2002Boskoff and Horja, 1994;Gehring and Hag, 2000;Hästö, 2003aHästö, , 2003bHästö, , 2004aHästö, , 2004bHästö, , 2006Hästö and Ibragimov, 2007;Hästö and Lindén, 2004;Ibragimov, 2002Ibragimov, , 2003aIbragimov, , 2003band Souza, 1999). Barbilian had a profound impact on 20th century Romanian arts and culture through his contributions in literature and mathematics.…”

The Apollonian Inner Metric

“…In [13,Theorem 2] he showed that for the Apollonian metric either none, one or all points are points of conformality. The middle case provides an interesting link to convex sets of constant width, see [13,Theorem 3]. We say that a domain G R n is quasi-isotropic if (G, α G ) is, similarly for isotropic.…”