The Lebesgue universal covering problem
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: Pál, Sprague and Hansen h...
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| Published in: | Journal of computational geometry Vol. 6; no. 1 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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Carleton University
01-09-2015
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| 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: Pál, 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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| AbstractList | 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: Pál, 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 $2.2 \cdot 10^{-5}$. |
| Author | Karine Bagdasaryan Philip Gibbs John C. Baez |
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| Snippet | 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... |
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| Title | The Lebesgue universal covering problem |
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