  We consider a Brownian particle with diffusion coefficient $D$ in a
$d$-dimensional ball of radius $R$ with reflecting boundaries. We study the
maximum $M_x(t)$ of the trajectory of the particle along the $x$-direction at
time $t$. In the long time limit, the maximum converges to the radius of the
ball $M_x(t) \to R$ for $t\to \infty$. We investigate how this limit is
approached and obtain an exact analytical expression for the distribution of
the fluctuations $\Delta(t) = [R-M_x(t)]/R$ in the limit of large $t$ in all
dimensions. We find that the distribution of $\Delta(t)$ exhibits a rich
variety of behaviors depending on the dimension $d$. These results are obtained
by establishing a connection between this problem and the narrow escape time
problem. We apply our results in $d=2$ to study the convex hull of the
trajectory of the particle in a disk of radius $R$ with reflecting boundaries.
We find that the mean perimeter $\langle L(t)\rangle$ of the convex hull
exhibits a slow convergence towards the perimeter of the circle $2\pi R$ with a
stretched exponential decay $2\pi R-\langle L(t)\rangle \propto
\sqrt{R}(Dt)^{1/4} \,e^{-2\sqrt{2Dt}/R}$. Finally, we generalise our results to
other confining geometries, such as the ellipse with reflecting boundaries. Our
results are corroborated by thorough numerical simulations.
