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Optimal $L^p$-approximation of convex sets by convex subsets

Zakaria Fattah, Ilias Ftouhi, Enrique Zuazua

TL;DR

The paper studies approximating a convex container $\Omega$ by convex subsets $\omega$ of fixed measure through $L^p$-type discrepancies between their support functions. It proves existence of minimizers for all $p\ge1$ and establishes $\Gamma$-convergence to the Hausdorff-distance problem as $p\to\infty$, linking smooth and non-smooth geometry in a unified framework. In 2D, the free boundary of optimal shapes is proven to be polygonal, and the authors develop two numerical schemes—Fourier-based and a convexity-parametrization approach—to accurately compute these shapes, with the latter showing superior performance in capturing boundary segments. The results illuminate the structure of optimal convex approximations and provide practical, robust computational tools for shape optimization under convexity and area constraints, with implications for geometric design and PDE-informed actuator/sensor placement.

Abstract

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.

Optimal $L^p$-approximation of convex sets by convex subsets

TL;DR

The paper studies approximating a convex container by convex subsets of fixed measure through -type discrepancies between their support functions. It proves existence of minimizers for all and establishes -convergence to the Hausdorff-distance problem as , linking smooth and non-smooth geometry in a unified framework. In 2D, the free boundary of optimal shapes is proven to be polygonal, and the authors develop two numerical schemes—Fourier-based and a convexity-parametrization approach—to accurately compute these shapes, with the latter showing superior performance in capturing boundary segments. The results illuminate the structure of optimal convex approximations and provide practical, robust computational tools for shape optimization under convexity and area constraints, with implications for geometric design and PDE-informed actuator/sensor placement.

Abstract

Given a convex set of , we consider the shape optimization problem of finding a convex subset , of a given measure, minimizing the -distance functional where and and are the support functions of and the fixed container , respectively. We prove the existence of solutions and show that this minimization problem -converges, when tends to , towards the problem of finding a convex subset , of a given measure, minimizing the Hausdorff distance to the convex . 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 , are given by polygonal lines. Still in the 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.
Paper Structure (18 sections, 14 theorems, 47 equations, 5 figures, 4 tables)

This paper contains 18 sections, 14 theorems, 47 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction and main results
  2. Notations and useful results
  3. Notations
  4. Properties of support functions in the planar case
  5. Proofs of Theorems \ref{['th:main_gamma']}, \ref{['th:main_gamma_2']} and \ref{['th:main_equivalence']}
  6. Proof of Theorem \ref{['th:main_gamma']}
  7. Proof of Theorem \ref{['th:main_gamma_2']}
  8. Proof of Theorem \ref{['th:main_equivalence']}
  9. Some remarks on the extremal cases $p=1$ and $p=+\infty$
  10. The case $p=1$
  11. The case $p=+\infty$
  12. Numerical simulations
  13. Method 1: Optimizing the Fourier coefficients
  14. Method 2: A new method based on a rigorous convexity parametrization
  15. Computational experiments
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\Omega\subset\mathbb{R}^n$ be a convex body, $p\in[1,+\infty)$ and $c\in[0,|\Omega|]$.

Figures (5)

  • Figure 1: The support function of the convex $\Omega$.
  • Figure 2: Parallel chord movements increase the perimeter while preserving the area.
  • Figure 3: The triangle $MAB$ is conjectured to be the solution of the reverse isoperimetric problem when the contained $\Omega$ is the triangle $ABC$.
  • Figure 4: First example for the history of convergence of the two methods (with a zoom in the figure in right).
  • Figure 5: Second example for the history of convergence of the two methods (with a zoom in the figure in right).

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Theorem 5
  • proof
  • Proposition 6
  • proof
  • Theorem 7
  • proof
  • ...and 20 more