Lebesgue泛覆盖问题
The Lebesgue Universal Covering Problem
摘要：
在1914年勒贝格定义了一个“普遍覆盖”是一个凸子集的平面，其中包含等距副本的任何子集的直径1。他的挑战是找到一个具有最小可能面积的泛覆盖，这一挑战已经被许多数学家所解决：爱因斯坦、斯普拉格和汉森都通过从以前已知的区域中删除区域来创建一个更小的泛覆盖。然而，Hansen的最后一次缩小是显微镜下的：他声称切除了一个$6 \cdot 10^{-18}$区域，但我们发现他实际上只切除了一个$8 \cdot 10^{-21}$区域。下面，在Greg Egan的帮助下，我们找到了一个面积小于$0.8441153$的新的更小的泛覆盖。这将之前最佳通用覆盖的面积减少了一个巨大的$2.2 \cdot 10^{-5}$。
英文摘要：
In 1914 Lebesgue defined a "universal covering" to be a convex subset of the plane that contains an isometric copy of any subset of diameter 1. His challenge of finding a universal covering with the least possible area has been addressed by various mathematicians: Pal, Sprague and Hansen have each created a smaller universal covering by removing regions from those known before. However, Hansen's last reduction was microsopic: he claimed to remove an area of $6 \cdot 10^{-18}$, but we show that he actually removed an area of just $8 \cdot 10^{-21}$. In the following, with the help of Greg Egan, we find a new, smaller universal covering with area less than $0.8441153$. This reduces the area of the previous best universal covering by a whopping $2.2 \cdot 10^{-5}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
