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 pn 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), d2r/dφ2 = f(φ), where f is Gaussian noise. On the basis of this relation we derive the asymptotic expansion log pn = –2n log n + n log(2π2e2)–c0n1/5 + · · ·, 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 1/5 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 δn of closest approach to the edge is exponentially distributed with average (2n)−1