Abstract
We give an affirmative answer to a question asked by Gardner and Mauldin (Geom. Dedicata 26, 323–332, 1988) about bijections of the plane taking each polygon with n sides onto a polygon with n sides. We also state and prove more general results in this spirit. For example, we show that an injective mapping taking each convex n-gon onto a non-degenerate n-gon (not necessarily convex or even simple) must be affine.
Mathematical Subject Classifications (2010): 52A10, 52A25
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Notes
- 1.
Alternatively, this can be easily seen by considering its convex hull whose boundary is a convex polygon with at least 3 corners and observing that all its corners are convex corners of the original polygon.
References
S. Artstein-Avidan, B.A. Slomka, A new fundamental theorem of affine geometry and applications, preprint
R.J. Gardner, R.D. Mauldin, Bijections of ℝ n onto itself. Geom. Dedicata 26, 323–332 (1988)
A.S. Jones, An elementary theorem for simple polygons. Math. Gaz. 59, 266–268 (1975)
Acknowledgements
The author would like to thank his advisor, Prof. Shiri Artstein–Avidan, for bringing the problems presented in [2] to his attention and for fruitful discussions. The author would also like to thank the anonymous referee for numerous helpful comments. This work has been partially supported by ISF grant No. 247/11.
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Slomka, B.A. (2013). On Polygons and Injective Mappings of the Plane. In: Ludwig, M., Milman, V., Pestov, V., Tomczak-Jaegermann, N. (eds) Asymptotic Geometric Analysis. Fields Institute Communications, vol 68. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6406-8_14
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DOI: https://doi.org/10.1007/978-1-4614-6406-8_14
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