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Observations on circumcenters in normed planes. (English) Zbl 1387.52005

Let \(E\) be a normed (or Minkowski) space of dimension 2. A triangle is a set \(T = \{a,b,c\}\) of three non collinear points of \(E\). A point \(x \in E\) is called a circumcenter of \(T\) if \(\| x -a \| = \| x -b \| = \| x -c \|\). The author shows that a triangle has a circumcenter if and only if the Birkhoff normals of the three line segments \([a,b], [b,c]\) and \([a,c]\) are not all parallel. As an application of this result he gives simplified proofs of some known theorems in the field.

MSC:

52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
46B20 Geometry and structure of normed linear spaces
51B20 Minkowski geometries in nonlinear incidence geometry
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References:

[1] Alonso, J., Martini, H., Spirova, M.: Minimal enclosing discs, circumcircles, and circumcenters on normed planes (part II). Comput. Geom. 45, 350-369 (2012) · Zbl 1251.65016 · doi:10.1016/j.comgeo.2012.02.003
[2] Jahn, T., Spirova, M.: On bisectors in normed spaces. Contrib. Discrete Math. 10(2), 1-9 (2015) · Zbl 1352.46015
[3] Kramer, H., Németh, A.B.: The application of Brouwer’s fixed point theorem of convex bodies (Romanian). An. Univ. Timisoara Ser. Sti. Mat. 13, 33-39 (1977) · Zbl 0449.52003
[4] Martini, H., Swanepoel, K.J., Weiss, G.: The geometry of Minkowski spaces—a survey, part I. Expo. Math. 19, 97-142 (2001) · Zbl 0984.52004 · doi:10.1016/S0723-0869(01)80025-6
[5] Martini, H., Swanepoel, K.J.: The geometry of Minkowski spaces—a survey, part II. Expo. Math. 22, 93-144 (2004) · Zbl 1080.52005 · doi:10.1016/S0723-0869(04)80009-4
[6] Mayer, A.E.: Eine Überkonvexität. Math. Z. 39, 511-531 (1935) · JFM 61.1427.02 · doi:10.1007/BF01201371
[7] Newman, M.H.A.: Elements of the Topology of Plane Sets of Points. Cambridge University Press, Cambridge (1961)
[8] Thompson, A.C.: Minkowski Geometry. Cambridge University Press, Cambridge (1996) · Zbl 0868.52001 · doi:10.1017/CBO9781107325845
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