Concepts of curvatures in normed planes |
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Authors: | Vitor Balestro Horst Martini Emad Shonoda |
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Institution: | 1. CEFET/RJ Campus Nova Friburgo, 28635000 Nova Friburgo - RJ, Brazil;2. Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, Germany;3. Department of Mathematics & Computer Science, Faculty of Science, Port Said University, 42521 Port Said, Egypt;4. Department of Mathematics, Mt. San Jacinto College, 92583 CA, United States |
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Abstract: | The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curvature concepts, we try to fill this gap. To complete and clarify the whole picture, we show which known concepts are equivalent, and add also a new type of curvature. Certainly, this yields a basis for further research and also for possible extensions of the whole existing framework. In addition, we derive various new results referring in full broadness to the variety of known curvature types in normed planes. These new results involve characterizations of curves of constant curvature, new characterizations of Radon planes and the Euclidean subcase, and analogues to classical statements like the four vertex theorem and the fundamental theorem on planar curves. We also introduce a new curvature type, for which we verify corresponding properties. As applications of the little theory developed in our expository paper, we study the curvature behavior of curves of constant width and obtain also new results on notions like evolutes, involutes, and parallel curves. |
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Keywords: | Corresponding author primary 52A10 secondary 26B15 46B20 51M25 51N25 52A10 52A21 53A04 53A35 Anti-norm Birkhoff orthogonality Evolute Four vertex theorem Minkowski curvature Normed plane |
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