Convex Hull of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math>Planar Brownian Motions: Exact Results and an Application to Ecology

Abstract

We compute exactly the mean perimeter and area of the convex hull of N independent planar Brownian paths each of duration T, both for open and closed paths. We show that the mean perimeter <L(N)>=alpha N sqrt[T] and the mean area <A(N)>=beta(N)T for all T. The prefactors alpha N and beta N, computed exactly for all N, increase very slowly (logarithmically) with increasing N. This slow growth is a consequence of extreme value statistics and has interesting implications in an ecological context in estimating the home range of a herd of animals with a population size N.