Finding the convex hull of a set using the flow by minimal curvature with an obstacle. A game theoretical approach

TL;DR

This paper introduces a game-theoretic method to compute the convex hull of a set by analyzing the long-term behavior of a minimal curvature flow with obstacles, extending previous mean curvature flow techniques.

Contribution

It develops a novel game-theoretic approximation for a minimal curvature flow with obstacles, enabling convex hull computation in higher dimensions.

Findings

Convergence of the flow to the convex hull as time approaches infinity.

Extension of game-theoretic methods from mean curvature flow to minimal curvature flow.

Validation of the approach through theoretical proofs and analysis.

Abstract

In this paper we look for the convex hull of a set using the geometric evolution by minimal curvature of a hypersurface that surrounds the set. To find the convex hull, we study the large time behavior of solutions to an obstacle problem for the level set formulation of the geometric flow driven by the minimum of the principal curvatures (that coincides with the mean curvature flow only in two dimensions). We prove that the superlevel set where the solution to this obstacle problem is positive converges as time goes to infinity to the convex hull of the obstacle. Our approach is based on a game-theoretic approximation for this geometric flow that is inspired by previous results for the mean curvature flow.