A trichotomy for rectangles inscribed in Jordan loops
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A trichotomy for rectangles inscribed in Jordan loops Richard Evan Schwartz1 Received: 8 June 2018 / Accepted: 13 February 2020 © Springer Nature B.V. 2020
Abstract We prove a general structural theorem about rectangles inscribed in Jordan loops. One corollary is that all but at most 4 points of any Jordan loop are vertices of inscribed rectangles. Another corollary is that a Jordan loop has an inscribed rectangle of every aspect ratio provided it has 3 points which are not vertices of inscribed rectangles. Keywords Square peg conjecture · Inscribed square · Inscribed rectangle · Configuration space Mathematics Subject Classification 51F99
1 Introduction A Jordan loop is the image of the circle under a continuous injective map into the plane. O. Toeplitz conjectured in 1911 that every Jordan loop contains 4 points which are the vertices of a square. This is often called the Square Peg Conjecture. An affirmative answer is known in many special cases. In 1913, Emch [4] proved the result for convex curves. In 1944, Shnirlmann [16] proved the result for sufficiently smooth curves. In 1961, Jerrald [6] extended this to the case of C 1 curves. For related work on the square peg problem, see [3]. Recently Tao [18] proved the result for special curves having even lower regularity. The above is a very partial survey of the literature. The 2014 survey paper by Matschke [9] and the recent book by Pak [14] have extensive discussions of the history of the Square Peg Conjecture and many additional references. There is also some work done in the case of rectangles. In 1977, Vaughan [20] gave a proof that every Jordan loop has an inscribed rectangle. A recent paper of Hugelmeyer [5] combines Vaughan’s basic idea with some very modern knot theory results to show that a √ smooth Jordan loop always has an inscribed rectangle of aspect ratio 3. The recent paper [1] proves that any quadrilateral inscribed in a circle can (up to similarity) be inscribed in any convex smooth curve. See also [10]. In the recent paper [2], the authors show that every
Richard Evan Schwartz Supported by N.S.F. Research Grant DMS-1204471.
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Richard Evan Schwartz [email protected] Brown University, Providence, USA
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Geometriae Dedicata
Jordan Loop contains a dense set of points which are vertices of inscribed rectangles. For additional work on inscribed rectangles, see [7,8,13]. Relatedly, one can consider the situation for triangles. In 1980, Meyerson [11] proved that all but at most 2 points of any Jordan loop are vertices of inscribed equilateral triangles. This result is sharp because two points of a suitable isosceles triangle are not vertices of inscribed equilateral triangles. In 1992, Neilson [12] proved that an arbitrary Jordan loop contains a dense set of points which are vertices of inscribed triangles of any given shape. We are going to prove a strong version of Meyerson’s Theorem for rectangles. Let I (γ ) denote the space of all labeled rectangles inscribed in γ . We always label a rectangle R so that the vertices of R go counterclo
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