内接于Jordan环的矩形的三分法
A Trichotomy for Rectangles Inscribed in Jordan Loops
摘要：
设g是任意Jordan环，G表示内接于g的矩形空间R，使得R的顶点的循环序无论是由R还是由g诱导的都是相同的。我们证明了G包含一个连通集S，满足以下三个性质之一：1。S由面积均匀的矩形组成，包括一个正方形，g的每个点都是S中矩形的顶点。2. S由具有所有可能的长宽比的矩形组成，并且g的所有点（除了最多4个点）都是S中矩形的顶点。3. S包含每一个足够小直径的矩形，并且g中除了至多2个点之外的所有点都是S中矩形的顶点。
英文摘要：
Let g be an arbitrary Jordan loop and let G denote the space of rectangles R which are inscribed in g in such a way that the cyclic order of the vertices of R is the same whether it is induced by R or by g. We prove that G contains a connected set S satisfying one of three properties: 1. S consists of rectangles of uniformly large area, including a square, and every point of g is the vertex of a rectangle in S. 2. S consists of rectangles having all possible aspect ratios, and all but at most 4 points of g are vertices of rectangles in S. 3. S contains rectangles of every sufficiently small diameter, and all but at most 2 points of g are vertices of rectangles in S.
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