Convex hull of a Brownian motion in confinement

We study the effect of confinement on the mean perimeter of the convex hull of a planar Brownian motion, defined as the minimum convex polygon enclosing the trajectory. We use a minimal model where an infinite reflecting wall confines the walk to one side. We show that the mean perimeter displays a surprising minimum with respect to the starting distance to the wall and exhibits a nonanalyticity for small distances. In addition, the mean span of the trajectory in a fixed direction θ∈]0,π/2[, which can be shown to yield the mean perimeter by integration over θ, presents these same two characteristics. This is in striking contrast to the one-dimensional case, where the mean span is an increasing analytical function. The nonmonotonicity in the two-dimensional case originates from the competition between two antagonistic effects due to the presence of the wall: reduction of the space accessible to the Brownian motion and effective repulsion.