Mathematika

Research Article

Inscribed squares and square-like quadrilaterals in closed curves

Walter Stromquista1

a1 Daniel H. Wagner, Associates, Station Square Two, Paoli, PA 19301, U.S.A..

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.

(Received November 29 1988)

Key Words:

  • 55M99: ALGEBRAIC TOPOLOGY; Classical topics.