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.
|Title of host publication||Asymptotic Geometric Analysis|
|Subtitle of host publication||Proceedings of the Fall 2010 Fields Institute Thematic Program|
|Editors||Monika Ludwig, Vladimir Pestov, Vitali Milman, Nicole Tomczak-Jaegermann|
|Number of pages||14|
|State||Published - 2013|
|Name||Fields Institute Communications|
Bibliographical noteFunding Information:
The author would like to thank his advisor, Prof. Shiri Artstein–Avidan, for bringing the problems presented in  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.
- Fundamental theorem of affine geometry
- Injective maps