Sylvester’s question and the random acceleration process

Abstract

Let n points be chosen randomly and independently in the unit disk. ‘Sylvester’s question’ concerns the probabilitypn that they are thevertices of a convex n-sided polygon. Here we establish the link with another problem. We show that for largen this polygon, when suitably parameterized by a functionr(ϕ) of thepolar 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 expansionlog 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 inn of the third term is a new result. The value of the exponent follows from recent work on the RAP due toGyörgyi et al (2007 Phys. Rev. E 75 021123). We show that then-sided polygon is effectively contained in an annulus of width∼n−4/5 along the edge of thedisk. The distance δn of closest approach to the edge is exponentially distributed with average(2n)−1.