The Lebesgue universal covering problem

Q4 Mathematics

International Journal of Computational Geometry & Applications Pub Date : 2015-02-04 DOI:10.20382/jocg.v6i1a12

J. Baez, Karine Bagdasaryan, P. Gibbs

引用次数: 10

Abstract

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 $2.2 \cdot 10^{-5}$.

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勒贝格普遍覆盖问题

1914年，勒贝格定义了一个“泛覆盖”，它是平面的一个凸子集，它包含任何直径为1的子集的等距副本。他的挑战是用最小可能的区域找到一个通用覆盖，这已经被许多数学家解决了:Pal, Sprague和Hansen都通过从已知的区域中移除区域来创建一个更小的通用覆盖。然而，Hansen的最后一次减少是微观的:他声称删除了$6 \cdot 10^{-18}$的面积，但我们表明他实际上只删除了$8 \cdot 10^{-21}$的面积。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。这将之前的最佳通用覆盖面积减少了$2.2 \cdot 10^{-5}$。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal of Computational Geometry & Applications 数学-计算机：理论方法

CiteScore

0.80

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The International Journal of Computational Geometry & Applications (IJCGA) is a quarterly journal devoted to the field of computational geometry within the framework of design and analysis of algorithms. Emphasis is placed on the computational aspects of geometric problems that arise in various fields of science and engineering including computer-aided geometry design (CAGD), computer graphics, constructive solid geometry (CSG), operations research, pattern recognition, robotics, solid modelling, VLSI routing/layout, and others. Research contributions ranging from theoretical results in algorithm design — sequential or parallel, probabilistic or randomized algorithms — to applications in the above-mentioned areas are welcome. Research findings or experiences in the implementations of geometric algorithms, such as numerical stability, and papers with a geometric flavour related to algorithms or the application areas of computational geometry are also welcome.