Sylvester’s question and the random acceleration process

Let n points be chosen randomly and independently in the unit disk. 'Sylvester's question' concerns the probability p n that they are the vertices of a convex n -sided polygon. Here we establish the link with another problem. We show that for large n this polygon, when suitably parameterized by a function r (ϕ) of the polar angle ϕ, satisfies the equation of the random acceleration process (RAP), d 2 r /dϕ 2 = f (ϕ), where f is Gaussian noise. On the basis of this relation we derive the asymptotic expansion log p n = –2 n log n + n log(2π 2 e 2 )– c 0 n 1/5 + · · ·, of which the first two terms agree with a rigorous result due to Bárány. The non-analyticity in n of the third term is a new result. The value of the exponent follows from recent work on the RAP due to Györgyi et al (2007 Phys. Rev. E 75 021123). We show that the n -sided polygon is effectively contained in an annulus of width ∼ n −4/5 along the edge of the disk. The distance δ n of closest approach to the edge is exponentially distributed with average (2 n ) −1 .