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In the April 1884 edition of The Educational Times, Sylvester
asked: what is the probability that four points chosen at random in the
plane form a convex quadrilateral? Cayley and de Morgan submitted
"solutions": 1/4 and 1/2, respectively. It is not surprising that no one
could detect which answer was correct, since this was asked decades before
the development of measure theory. Sylvester himself declared: "The problem
does not admit a determinate solution". Sylvester later refined his
question, but even today some very basic problems around Sylvester's
question remain open.
Let R be a closed, bounded region in the plane, and let q(R) be the
probability that four points chosen at random from R define a convex
quadrilateral. It is easy to see that q(R) can be made arbitrarily close to
1 (make R a very thin annulus). Thus it remains to consider the infimum q*
of q(R). The exact determination of q* (known as Sylvester's Four Point
Constant) is still an open problem. Scheinerman and Wilf revelead the close
connection between q* and a classical combinatorial geometry problem, the
rectilinear crossing number of the complete graph (that is, the minimum
number of crossings in a drawing of the complete graph in the plane in which
edges are straight segments). Around 2000, the ratio between the best lower
and upper bounds known for q* was around 0.755. Nowadays, it's greater than
0.998. Are we really in the verge of determining Sylvester's Four Point
Constant? In this talk, we will survey the history of Sylvester's Four Point
Problem, and we will also give an overview of the recent developments in the
chase for the exact value of q*.
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