  We consider the set of points chosen randomly, independently and uniformly in
the $d$-dimensional spherical layer. A set of points is called $1$-convex if
all its points are vertices of the convex hull of this set. In \cite{3} an
estimate for the cardinality of the set of points for which this set is
$1$-convex with probability close to $1$ was obtained. In this paper we obtain
an improved estimate.
