  The purpose of this paper is to highlight some hidden Markovian structure of
the concave majorant of the Brownian motion. Several distributional identities
are implied by the joint law of a standard one-dimensional Brownian motion $B$
and its almost surely unique concave majorant $K$ on $[0,\infty)$. In
particular, the one-dimensional distribution of $2 K_t - B_t$ is that of
$R_5(t)$, where $R_5$ is a $5-$dimensional Bessel process with $R_5(0) = 0$.
The process $2K-B$ shares a number of other properties with $R_5$, and we
conjecture that it may have the distribution of $R_5$. We also describe the
distribution of the convex minorant of a three-dimensional Bessel process with
drift.
