On the Square Peg Problem and some Relatives

Details On the Square Peg Problem and some Relatives On the Square Peg Problem and some Relatives

2009-12-31

arxiv.org/abs/1001.0186v1

Mathematics

Metric Geometry

15 pages, 14 figures

Scientific paper

The Square Peg Problem asks whether every continuous simple closed planar curve contains the four vertices of a square. This paper proves this for the largest so far known class of curves. Furthermore we solve an analogous Triangular Peg Problem affirmatively, state topological intuition why the Rectangular Peg Problem should hold true, and give a fruitful existence lemma of edge-regular polygons on curves. Finally, we show that the problem of finding a regular octahedron on embedded spheres in R^3 has a "topological counter-example", that is, a certain test map with boundary condition exists.

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