Hamilton-Jacobi scaling limits of Pareto peeling in 2D

TL;DR

This paper establishes that Pareto hull peeling in 2D converges to a solution of a Hamilton-Jacobi equation, revealing a new scaling limit distinct from convex peeling's curvature flow.

Contribution

It proves the scaling limit of Pareto peeling in 2D is governed by a Hamilton-Jacobi equation with an explicit non-coercive, non-convex Hamiltonian, contrasting with convex peeling.

Findings

Pareto peeling converges to a Hamilton-Jacobi PDE in 2D.

The limiting Hamiltonian is explicitly characterized as non-coercive and non-convex.

Contrast with convex peeling, which converges to curvature flow.

Abstract

Pareto hull peeling is a discrete algorithm, generalizing convex hull peeling, for sorting points in Euclidean space. We prove that Pareto peeling of a random point set in two dimensions has a scaling limit described by a first-order Hamilton-Jacobi equation and give an explicit formula for the limiting Hamiltonian, which is both non-coercive and non-convex. This contrasts with convex peeling, which converges to curvature flow. The proof involves direct geometric manipulations in the same spirit as Calder (2016).