  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 its one side. We show that the mean perimeter displays a
surprising minimum with respect to the starting distance to the wall and
exhibits a non-analyticity for small distances. In addition, the mean span of
the trajectory in a fixed direction {$\theta \in ]0,\pi/2[$}, which can be
shown to yield the mean perimeter by integration over $\theta$, presents these
same two characteristics. This is in striking contrast with the one dimensional
case, where the mean span is an increasing analytical function. The
non-monotonicity in the 2D 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.
