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ANY CYCLIC QUADRILATERAL CAN BE INSCRIBED IN ANY CLOSED CONVEX SMOOTH CURVE

Part of: Classical differential geometry Fiber spaces and bundles Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  31 May 2018

ARSENIY AKOPYAN  and
SERGEY AVVAKUMOV
ARSENIY AKOPYAN
Affiliation:
Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria; [email protected], [email protected]
SERGEY AVVAKUMOV
Affiliation:
Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria; [email protected], [email protected]

Abstract

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We prove that any cyclic quadrilateral can be inscribed in any closed convex $C^{1}$-curve. The smoothness condition is not required if the quadrilateral is a rectangle.

MSC classification

Primary: 53A04: Curves in Euclidean space 55M20: Fixed points and coincidences
Secondary: 55R80: Discriminantal varieties, configuration spaces
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 6 , 2018 , e7
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2018

References

Emch, A., ‘Some properties of closed convex curves in a plane’, Amer. J. Math. 35(4) (1913), 407–412.CrossRefGoogle Scholar
Emch, A., ‘On some properties of the medians of closed continuous curves formed by analytic arcs’, Amer. J. Math. 38(1) (1916), 6–18.Google Scholar
Griffiths, H. B., ‘The topology of square pegs in round holes’, Proc. London Math. Soc. (3) 62(3) (1991), 647–672.CrossRefGoogle Scholar
Gromov, M. L., ‘Simplexes inscribed on a hypersurface’, Mat. Zametki 5 (1969), 81–89.Google Scholar
Hausel, T., Makai, E. and Szűcs, A., ‘Inscribing cubes and covering by rhombic dodecahedra via equivariant topology’, Mathematika 47(1–2) (2000), 371–397.Google Scholar
Karasev, R. N., ‘Inscribing a regular crosspolytope’. Preprint (2009), arXiv:0905.2671.Google Scholar
Karasev, R. N., ‘A note on Makeev’s conjectures’, J. Math. Sci. 212(5) (2016), 521–526.Google Scholar
Makeev, V. V., ‘Quadrangles inscribed in a closed curve’, Math. Notes 57(1) (1995), 91–93.Google Scholar
Makeev, V. V., ‘Universally inscribed and outscribed polytopes’, Doctor of Mathematics Thesis, Saint-Petersburg State University, 2003.Google Scholar
Matschke, B., ‘Equivariant topology methods in discrete geometry’, PhD Thesis, Freie Universität Berlin, 2011.Google Scholar
Matschke, B., ‘A survey on the square peg problem’, Notices Amer. Math. Soc. 61(4) (2014), 346–352.Google Scholar
Matschke, B., ‘Quadrilaterals inscribed in convex curves’. Preprint (2018),arXiv:1801.01945.Google Scholar
Nielsen, M. J. and Wright, S. E., ‘Rectangles inscribed in symmetric continua’, Geom. Dedicata 56(3) (1995), 285–297.CrossRefGoogle Scholar
Pak, I., ‘The discrete square peg problem’. Preprint (2008), arXiv:0804.0657.Google Scholar
Stromquist, W., ‘Inscribed squares and square-like quadrilaterals in closed curves’, Mathematika 36(2) (1990), 187–197; (1989).Google Scholar
Šnirel’man, L. G., ‘On certain geometrical properties of closed curves’, Uspekhi Mat. Nauk 10 (1944), 34–44.Google Scholar
Tao, T., ‘An integration approach to the Toeplitz square peg problem’, Forum Math. Sigma 5 (2017).Google Scholar