The limit shape of convex hull peeling

TL;DR

This paper establishes that convex hull peeling of random points in higher dimensions approximates a geometric motion governed by Gaussian curvature, using advanced PDE and probabilistic methods.

Contribution

It introduces a novel connection between convex peeling and curvature-driven geometric motions, employing viscosity solutions and martingale techniques.

Findings

Convex peeling approximates motion by Gaussian curvature in high dimensions.

The use of viscosity solutions provides a new interpretation of the limiting PDE.

Martingale methods solve the associated cell problem effectively.

Abstract

We prove that the convex peeling of a random point set in dimension d approximates motion by the 1/(d + 1) power of Gaussian curvature. We use viscosity solution theory to interpret the limiting partial differential equation. We use the Martingale method to solve the cell problem associated to convex peeling. Our proof follows the program of Armstrong-Cardaliaguet for homogenization of geometric motions, but with completely different ingredients.