Finding the convex envelope of a boundary datum using random geometric graphs

TL;DR

This paper introduces a method to approximate the convex envelope of a boundary function within a domain using solutions to equations on random geometric graphs, demonstrating convergence as the number of points increases.

Contribution

It presents a novel approach connecting random geometric graphs with convex envelope approximation and proves convergence under certain conditions.

Findings

Solution on the graph converges to the convex envelope as points increase.

The method approximates the first eigenvalue of the Hessian using random graphs.

Convergence is established under specific assumptions on the connection radius.

Abstract

In this paper we approximate the convex envelope of a boundary datum inside a bounded domain in the Euclidean space. We work with a random graph that is obtained as random points with uniform distribution that are connected by proximity ($x\sim y$ when $|x-y|<r$). On the graph we solve an equation (that approximate the first eigenvalue of the Hessian of a smooth function) with an exterior datum. Under appropriate assumptions on $r$ we show that the unique solution to the equation in the graph converges to the convex envelope of the boundary datum as the number of points goes to infinity.