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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.

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