Some Remarks on the Visual Angle Metric
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Some Remarks on the Visual Angle Metric Parisa Hariri1 · Matti Vuorinen1 · Gendi Wang2
Received: 22 October 2014 / Revised: 3 June 2015 / Accepted: 20 June 2015 / Published online: 19 August 2015 © Springer-Verlag Berlin Heidelberg 2015
Abstract We show that the visual angle metric and the triangular ratio metric are comparable in convex domains. We also find the extremal points for the visual angle metric in the half space and in the ball by use of a construction based on hyperbolic geometry. Furthermore, we study distortion properties of quasiconformal maps with respect to the triangular ratio metric and the visual angle metric. Keywords
Triangular ratio metric · Visual angle metric · Quasiconformal maps
Mathematics Subject Classification
30C65 · 51M10
1 Introduction Geometric function theory studies classes of mappings between subdomains of the Euclidean space Rn , n ≥ 2. These classes include both injective and non-injective mappings. In particular, Lipschitz, quasiconformal, and quasiregular mappings along with their generalizations such as maps with integrable dilatation are in focus. On the
Communicated by Edward Crane.
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Parisa Hariri [email protected] Matti Vuorinen [email protected] Gendi Wang [email protected]
1
Department of Mathematics and Statistics, University of Turku, Turku, Finland
2
Department of Physics and Mathematics, University of Eastern Finland, Joensuu, Finland
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other hand, this theory has also been extended to Banach spaces and even to metric spaces. What is common to these theories is that various metrics are extensively used as powerful tools, e.g., Väisälä’s theory of quasiconformality in Banach spaces [14] is based on the study of metrics: the norm metric, the quasihyperbolic metric and the distance ratio metric. In recent years, several authors have studied the geometries defined by these and other related metrics [7,8,10,12,13]. For a survey of these topics the reader is referred to [17]. The main purpose of this paper is to continue the study of some of these metrics. For a domain G Rn and x, y ∈ G, the visual angle metric is defined by vG (x, y) = sup{(x, z, y) : z ∈ ∂G} ∈ [0, π ],
(1.1)
where ∂G is not a proper subset of a line. This metric was introduced and studied very recently in [11]. It is clear that a point z ∈ ∂G exists for which this supremum is attained, such a point z is called an extremal point for vG (x, y). For a domain G Rn and x, y ∈ G, the triangular ratio metric is defined by sG (x, y) = sup z∈∂G
|x − y| ∈ [0, 1]. |x − z| + |z − y|
(1.2)
Again, the existence of an extremal boundary point is obvious. This metric has been studied in [3,9]. The above two metrics are closely related, for instance, both depend on extremal boundary points. It is easy to see that both metrics are monotone with respect to domain. Thus, if D and G are domains in Rn and D ⊂ G then for all x, y ∈ D, we have s D (x, y) ≥ sG (x, y), v D (x, y) ≥ vG (x, y). On the other hand, we will see that these two metrics are not comparable in some domains
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