# The opaque square

• Published in 2013
In the collections
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.

### 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}
}