Optimal Lp-approximation of convex sets by convex subsets

Fattah Z, Ftouhi I, Zuazua Iriondo E (2025)

Publication Language: English

Publication Type: Journal article

Publication year: 2025

Journal

Nonlinear Analysis - Theory Methods & Applications Elsevier

Book Volume: 261

Article Number: 113866

DOI: 10.1016/j.na.2025.113866

Abstract

Given a convex set Ω of Rn, we consider the shape optimization problem of finding a convex subset ω⊂Ω, of a given measure, minimizing the p-distance functional [Formula presented] 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 Γ-converges, 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 2D 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 Bogosel (2023), allowing to obtain accurate results, as demonstrated through numerical experiments.

Authors with CRIS profile

Ilias Ftouhi Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur) Enrique Zuazua Iriondo Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

École Marocaine des Sciences de l'Ingénieur (EMSI) MA Morocco (MA)

How to cite

APA:

Fattah, Z., Ftouhi, I., & Zuazua Iriondo, E. (2025). Optimal Lp-approximation of convex sets by convex subsets. Nonlinear Analysis - Theory Methods & Applications, 261. https://doi.org/10.1016/j.na.2025.113866

MLA:

Fattah, Zakaria, Ilias Ftouhi, and Enrique Zuazua Iriondo. "Optimal Lp-approximation of convex sets by convex subsets." Nonlinear Analysis - Theory Methods & Applications 261 (2025).

BibTeX: Download