Optimal $L^p$-approximation of convex sets by convex subsets
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866929654572515328 |
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| author | Fattah, Zakaria Ftouhi, Ilias Zuazua, Enrique |
| author_facet | Fattah, Zakaria Ftouhi, Ilias Zuazua, Enrique |
| contents | Given a convex set $Ω$ of $\mathbb{R}^n$, we consider the shape optimization problem of finding a convex subset $ω\subset Ω$, of a given measure, minimizing the $p$-distance functional $$\mathcal{J}_p(ω) := \left(\int_{\mathbb{S}^{n-1}} |h_Ω-h_ω|^p d\mathcal{H}^{n-1}\right)^{\frac{1}{p}},$$ where $1 \le p <\infty$ and $h_ω$ and $h_Ω$ 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 $+\infty$, towards the problem of finding a convex subset $ω\subset Ω$, 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 recent numerical scheme, allowing to obtain accurate results, as demonstrated through numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00928 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal $L^p$-approximation of convex sets by convex subsets Fattah, Zakaria Ftouhi, Ilias Zuazua, Enrique Optimization and Control Given a convex set $Ω$ of $\mathbb{R}^n$, we consider the shape optimization problem of finding a convex subset $ω\subset Ω$, of a given measure, minimizing the $p$-distance functional $$\mathcal{J}_p(ω) := \left(\int_{\mathbb{S}^{n-1}} |h_Ω-h_ω|^p d\mathcal{H}^{n-1}\right)^{\frac{1}{p}},$$ where $1 \le p <\infty$ and $h_ω$ and $h_Ω$ 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 $+\infty$, towards the problem of finding a convex subset $ω\subset Ω$, 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 recent numerical scheme, allowing to obtain accurate results, as demonstrated through numerical experiments. |
| title | Optimal $L^p$-approximation of convex sets by convex subsets |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2501.00928 |