arXiv 1809.030702018-09math.MG

矩形重合和扫掠

Rectangle Coincidences and Sweepouts

Richard Evan Schwartz

备注：: 19 pages, traditional proof. This update corrects a glitch in which I left off the statement that the extremal chords need to be positively oriented, in a sense that is explained in the paper

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摘要：

证明了一个分段光滑环上内接矩形的连续路径的积分公式。然后，我们使用这个积分公式表明，（一个非常温和的通用性假设）矩形重合的数量，非正式地描述为内接矩形的数量减去内接矩形的等距类的数量，线性增长的数量正向极端弦-又名。多边形的直径

英文摘要：

We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We then use this integral formula to show that (with a very mild genericity hypothesis) the number of rectangle coincidences, informally described as the number of inscribed rectangles minus the number of isometry classes of inscribed rectangles, grows linearly with the number of positively oriented extremal chords -- a.k.a. diameters -- of the polygon

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