Polygons as optimal shapes with convexity constraint

TL;DR

This paper investigates shape optimization problems with convexity constraints, demonstrating that solutions are polygons for a broad class of functionals by deriving extremality conditions and analyzing their implications.

Contribution

It introduces a parameterization of convex domains and proves that solutions are polygons under certain concavity-like conditions on the functional.

Findings

Solutions are polygons for a large class of functionals.

Derived extremality conditions for convex shape optimization.

Provided a new characterization of optimal convex shapes.

Abstract

In this paper, we focus on the following general shape optimization problem: $$ \min\{J(\Om), \Om convex, \Om\in\mathcal S_{ad}\}, $$ where $\mathcal S_{ad}$ is a set of 2-dimensional admissible shapes and $J:\mathcal{S}_{ad}\to\R$ is a shape functional. Using a specific parameterization of the set of convex domains, we derive some extremality conditions (first and second order) for this kind of problem. Moreover, we use these optimality conditions to prove that, for a large class of functionals (satisfying a concavity like property), any solution to this shape optimization problem is a polygon.