We settle J. Wetzel’s 1970’s conjecture and show that a $$30^{\circ }$$ circular sector of unit radius can accommodate every planar arc of unit length. Leo Moser asked in 1966 for the (convex) region with the smallest area in the plane that can accommodate each arc of unit length. With area $$\pi /12,$$ this sector is the smallest such set presently known. Moser’s question has prompted a multitude of papers on related problems over the past 50 years, most remaining unanswered.

中文翻译：

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Wetzel 的扇区涵盖单位弧

我们解决了 J. Wetzel 1970 年的猜想，并表明单位半径的 $$30^{\circ }$$ 圆形扇区可以容纳单位长度的每个平面弧。Leo Moser 于 1966 年要求平面中面积最小的（凸）区域可以容纳单位长度的每个弧。面积 $$\pi /12,$$ 这个扇区是目前已知的最小的这样的集合。在过去的 50 年里，莫泽的问题引发了大量关于相关问题的论文，其中大部分仍未得到解答。