Interesting Esoterica

The opaque square

Article by Adrian Dumitrescu and Minghui Jiang
  • Published in 2013
  • Added on
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The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916. We call such a set {\em opaque} or a {\em barrier} for the square. The shortest known barrier has length $\sqrt{2}+ \frac{\sqrt{6}}{2}= 2.6389\ldots$. The current best lower bound for the length of a (not necessarily connected) barrier is $2$, as established by Jones about 50 years ago. No better lower bound is known even if the barrier is restricted to lie in the square or in its close vicinity. Under a suitable locality assumption, we replace this lower bound by $2+10^{-12}$, which represents the first, albeit small, step in a long time toward finding the length of the shortest barrier. A sharper bound is obtained for interior barriers: the length of any interior barrier for the unit square is at least $2 + 10^{-5}$. Two of the key elements in our proofs are: (i) formulas established by Sylvester for the measure of all lines that meet two disjoint planar convex bodies, and (ii) a procedure for detecting lines that are witness to the invalidity of a short bogus barrier for the square.

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BibTeX entry

@article{Theopaquesquare,
	title = {The opaque square},
	abstract = {The problem of finding small sets that block every line passing through a
unit square was first considered by Mazurkiewicz in 1916. We call such a set
{\{}\em opaque{\}} or a {\{}\em barrier{\}} for the square. The shortest known barrier has
length {\$}\sqrt{\{}2{\}}+ \frac{\{}\sqrt{\{}6{\}}{\}}{\{}2{\}}= 2.6389\ldots{\$}. The current best lower
bound for the length of a (not necessarily connected) barrier is {\$}2{\$}, as
established by Jones about 50 years ago. No better lower bound is known even if
the barrier is restricted to lie in the square or in its close vicinity. Under
a suitable locality assumption, we replace this lower bound by {\$}2+10^{\{}-12{\}}{\$},
which represents the first, albeit small, step in a long time toward finding
the length of the shortest barrier. A sharper bound is obtained for interior
barriers: the length of any interior barrier for the unit square is at least {\$}2
+ 10^{\{}-5{\}}{\$}. Two of the key elements in our proofs are: (i) formulas established
by Sylvester for the measure of all lines that meet two disjoint planar convex
bodies, and (ii) a procedure for detecting lines that are witness to the
invalidity of a short bogus barrier for the square.},
	url = {http://arxiv.org/abs/1311.3323v1 http://arxiv.org/pdf/1311.3323v1},
	author = {Adrian Dumitrescu and Minghui Jiang},
	comment = {},
	urldate = {2017-05-22},
	archivePrefix = {arXiv},
	eprint = {1311.3323},
	primaryClass = {math.CO},
	year = 2013,
	collections = {easily-explained}
}