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Inscribed squares and square-like quadrilaterals in closed curves

Part of: Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  26 February 2010

Walter Stromquist
Walter Stromquist
Affiliation:
Daniel H. Wagner, Associates, Station Square Two, Paoli, PA 19301, U.S.A..
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Abstract

We show that for every smooth curve in Rn, there is a quadrilateral with equal sides and equal diagonals whose vertices lie on the curve. In the case of a smooth plane curve, this implies that the curve admits an inscribed square, strengthening a theorem of Schnirelmann and Guggenheimer. “Smooth” means having a continuously turning tangent. We give a weaker condition which is still sufficient for the existence of an inscribed square in a plane curve, and which is satisfied if the curve is convex, if it is a polygon, or (with certain restrictions) if it is piecewise of class C1. For other curves, the question remains open.

MSC classification

Secondary: 55M99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 36 , Issue 2 , December 1989 , pp. 187 - 197
Copyright
Copyright © University College London 1989

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