# The opaque square

- 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,Geometry} }