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(phi) of the polar angle phi, satisfies the equation of the random acceleration process (RAP), d(2)r/d phi(2) = f(phi), where f is Gaussian noise. On the basis of this relation we derive the asymptotic expansion log p(n) = -2n log n+n log(2 pi(2)e(2))-c(0)n(1/5)+..., of which the first two terms agree with a rigorous result due to Barany. The non-analyticity in n of the third term is a new result. The value 1/5 of the exponent follows from recent work on the RAP due to Gyorgyi et al (2007 Phys. Rev. E 75 021123). We show that the n-sided polygon is effectively contained in an annulus of width similar to n(-4/5) along the edge of the disk. The distance delta(n) of closest approach to the edge is exponentially distributed with average (2n)(-1).