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 $N$.

• Received 2 July 2009

DOI:https://doi.org/10.1103/PhysRevLett.103.140602