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 2β+26ββ=2.6389β¦. 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+10β12,
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+10β5. 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.Comment: 23 pages, 8 figure