## Sylvester's Four Point Constant: closing in (or are we?)

Speaker: | Gelasio Salazar |
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Affiliation: | Universidad Autonoma de San Luis Potosi, Mexico |

Room: | Mathematics & Computer Building (MC) 5158 |

### Abstract:

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*.