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Inscribed squares and square-like quadrilaterals in closed curves

  • Walter Stromquist (a1)

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

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