Details

Snapshot 1: a convex quadrilateral: the convex hull has four vertices

Snapshot 2: a concave quadrilateral: the convex hull has only three vertices

This Demonstration illustrates the four-point problem of J. J. Sylvester (1814–1897). He presented the problem in the April 1864 issue of Educational Times; see [1], [2, pp. 15–21]. The problem was as follows: if we pick four points at random inside some given convex region, what is the probability that the four points are the vertices of a concave quadrilateral? Wilhelm Blaschke (1885–1962) showed that the probability is at least and at most , depending on the form of the convex region. W. S. B. Woolhouse (1809–1893) showed that the lower bound is achieved if the convex region is a circle; for a square, the probability is and for a regular hexagon, . The Demonstration is based on [2], where a method is presented to generate random points in a circle.

References

[1] R. E. Pfiefer, "The Historical Development of J. J. Sylvester's Four Point Problem," Mathematics Magazine, 62(5), 1989 pp. 309–317.

[2] P. J. Nahin, Digital Dice: Computational Solutions to Practical Probability Problems, Princeton, NJ: Princeton University Press, 2008.