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  • Cited by 4
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Makeev, V. V. 2011. Closed polylines inscribed in a closed space curve. Journal of Mathematical Sciences, Vol. 175, Issue. 5, p. 556.


    Sagols, Feliú and Marín, Raúl 2011. Two discrete versions of the Inscribed Square Conjecture and some related problems. Theoretical Computer Science, Vol. 412, Issue. 15, p. 1301.


    Vrećica, Siniša T. and Živaljević, Rade T. 2011. Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem. Israel Journal of Mathematics, Vol. 184, Issue. 1, p. 221.


    Nielsen, Mark J. 1995. Rhombi inscribed in simple closed curves. Geometriae Dedicata, Vol. 54, Issue. 3, p. 245.


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Inscribed squares and square-like quadrilaterals in closed curves

  • Walter Stromquist (a1)
  • DOI: http://dx.doi.org/10.1112/S0025579300013061
  • Published online: 01 February 2010
Abstract
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.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

1.Victor Klee . Some unsolved problems in plane geometry. Math. Magazine, 52 (1979), 131145.

3.A. Emch . Some properties of closed convex curves in a plane. Amer. J. of Math., 35 (1913), 407412.

4.A. Emch . On the medians of a closed convex polygon. Amer. J. of Math., 37 (1915), 1928.

6.H. Guggenheimer . Finite sets on curves and surfaces. Isreal J. Math., 3 (1965), 104112.

7.R. P. Jerrard . Inscribed squares in plane curves. Trans. Amer. Math. Soc, 98 (1961), 234241.

8.Roger Fenn . The table theorem. Bull. London Math. Soc, 2 (1970), 7376.

10.S. Lefschetz . Introduction to Topology (Princeton University Press, 1949).

11.M. Meyerson . Convexity and the table theorem. Pacific J. Math., 97 (1981), 167169.

12.M. Meyerson . Remarks on Fenn's “The Table Theorem” and Zaks' “The Chair Theorem”. Pacific J. Math., 110 (1984), 167169.

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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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