Convex hull of n planar Brownian paths: an exact formula for the average number of edges

TL;DR

This paper derives an exact formula for the average number of edges on the convex hull boundary of n independent planar Brownian paths, using a novel counting criterion and a mapping to constrained Brownian motions.

Contribution

It introduces a new exact formula for the average number of edges on the convex hull of multiple Brownian paths, advancing understanding of their geometric properties.

Findings

Derived an exact formula for the average number of edges.

Confirmed the formula by recovering known perimeter results.

Introduced a new counting criterion for small edges.

Abstract

We establish an exact formula for the average number of edges appearing on the boundary of the global convex hull of n independent Brownian paths in the plane. This requires the introduction of a counting criterion which amounts to "cutting off" edges that are, in a specific sense, small. The main argument consists in a mapping between planar Brownian convex hulls and configurations of constrained, independent linear Brownian motions. This new formula is confirmed by retrieving an existing exact result on the average perimeter of the boundary of Brownian convex hulls in the plane.