F. Zakaria, I. Ftouhi, E. Zuazua (2025) Optimal L^p-approximation of convex sets by convex subsets
Abstract. Given a convex set Ω of R n, we consider the shape optimization problem of finding a convex subset ω ⊂ Ω, of a given measure, minimizing the p-distance functional Jp(ω) := Z Sn−1 |hΩ − hω| p dHn−1 1 p , where 1 ≤ p < ∞ and hω and hΩ are the support functions of ω and the fixed container Ω, respectively. We prove the existence of solutions and show that this minimization problem Γconverges, when p tends to +∞, towards the problem of finding a convex subset ω ⊂ Ω, of a given measure, minimizing the Hausdorff distance to the convex Ω. In the planar case, we show that the free parts of the boundary of the optimal shapes, i.e., those that are in the interior of Ω, are given by polygonal lines. Still in the 2 − d setting, from a computational perspective, the classical method based on optimizing Fourier coefficients of support functions is not efficient, as it is unable to efficiently capture the presence of segments on the boundary of optimal shapes. We subsequently propose a method combining Fourier analysis and a numerical scheme recently introduced in [4], allowing to obtain accurate results, as demonstrated through numerical experiments. arxiv: 2501.00928