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

TL;DR

This paper improves estimates for the number of randomly chosen points in a spherical layer that form a 1-convex set, where all points are vertices of their convex hull, in high-dimensional spaces.

Contribution

The paper provides an improved estimate for the size of point sets that are 1-convex with high probability in a d-dimensional spherical layer.

Findings

Enhanced bounds for 1-convex point set cardinality

Probabilistic estimates for convex hull vertices in high dimensions

Refinement over previous estimates in spherical layers

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.