Abstract:We classify the set of quadrilaterals that can be inscribed in convex Jordan curves, in the continuous as well as in the smooth case. This answers a question of Makeev in the special case of convex curves. The difficulty of this problem comes from the fact that standard topological arguments to prove the existence of solutions do not apply here due to the lack of sufficient symmetry. Instead, the proof makes use of an area argument of Karasev and Tao, which we furthermore simplify and elaborate on. The continuous case requires an additional analysis of the singular points, and a small miracle, which then extends to show that the problems of inscribing isosceles trapezoids in smooth curves and in piecewise $C^1$ curves are equivalent.

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中文翻译：

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内接于凸曲线的四边形

摘要：我们对可以内接在凸乔丹曲线中的四边形集进行分类，在连续和光滑情况下。这在凸曲线的特殊情况下回答了 Makeev 的问题。这个问题的困难在于，由于缺乏足够的对称性，证明解存在的标准拓扑论证在这里不适用。相反，该证明使用了 Karasev 和 Tao 的区域论证，我们进一步简化和阐述了它们。连续的情况需要对奇异点进行额外的分析，以及一个小奇迹，然后扩展到表明在平滑曲线和分段 $C^1$ 曲线中内切等腰梯形的问题是等效的。

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