  We study various properties of the convex hull of a planar Brownian motion,
defined as the minimum convex polygon enclosing the trajectory, in the presence
of an infinite reflecting wall. Recently, in a Rapid Communication [Phys. Rev.
E \textbf{91}, 050104(R) (2015)], we announced that the mean perimeter of the
convex hull at time $t$, rescaled by $\sqrt{Dt}$, is a non-monotonous function
of the initial distance to the wall. In the present article, we first give all
the details of the derivation of this mean rescaled perimeter, in particular
its value when starting from the wall and near the wall. We then determine the
physical mechanism underlying this surprising non-monotonicity of the mean
rescaled perimeter by analyzing the impact of the wall on two complementary
parts of the convex hull. Finally, we provide a further quantification of the
convex hull by determining the mean length of the portion of the reflecting
wall visited by the Brownian motion as a function of the initial distance to
the wall.
