On Polygons and Injective Mappings of the Plane

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


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.

Original languageEnglish
Title of host publicationAsymptotic Geometric Analysis
Subtitle of host publicationProceedings of the Fall 2010 Fields Institute Thematic Program
EditorsMonika Ludwig, Vladimir Pestov, Vitali Milman, Nicole Tomczak-Jaegermann
Number of pages14
StatePublished - 2013
Externally publishedYes

Publication series

NameFields Institute Communications
ISSN (Print)1069-5265

Bibliographical note

Funding 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
  • Polygons


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