On the $1$-convexity of random points in the $d$-dimensional spherical layer

Abstract

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.