Polygons as optimal shapes with convexity constraint

Details Polygons as optimal shapes with convexity constraint Polygons as optimal shapes with convexity constraint

2009-02-18

arxiv.org/abs/0902.3062v1

Mathematics

Optimization and Control

Scientific paper

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.

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