Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology

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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 < Lₙ > = αₙ, √T and the mean area <Aₙ> = βₙ T for all T. The prefactors αₙ and βₙ, 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 implication in ecological context in estimating the home range of a herd of animals with population size N.