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

TL;DR

This paper studies the optimal convex subset approximation of a convex set in b^n using a shape optimization approach based on the p-distance functional, proving existence, convergence, and providing computational methods especially in 2D.

Contribution

It introduces a new shape optimization framework for convex set approximation, proves existence of solutions, analyzes bc-convergence, and develops an efficient computational method in 2D.

Findings

Existence of optimal convex subsets minimizing the p-distance functional.

bc-convergence of the minimization problem to Hausdorff distance minimization as p b7 f7.

In 2D, optimal shapes have polygonal free boundary parts, and the new computational method accurately captures boundary segments.

Abstract

Given a convex set $\Omega$ of $\mathbb{R}^n$, we consider the shape optimization problem of finding a convex subset $\omega\subset \Omega$, of a given measure, minimizing the $p$-distance functional $$\mathcal{J}_p(\omega) := \left(\int_{\mathbb{S}^{n-1}} |h_\Omega-h_\omega|^p d\mathcal{H}^{n-1}\right)^{\frac{1}{p}},$$ where $1 \le p <\infty$ and $h_\omega$ and $h_\Omega$ are the support functions of $\omega$ and the fixed container $\Omega$, respectively. We prove the existence of solutions and show that this minimization problem $\Gamma$-converges, when $p$ tends to $+\infty$, towards the problem of finding a convex subset $\omega\subset \Omega$, of a given measure, minimizing the Hausdorff distance to the convex $\Omega$. 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 $\Omega$, 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.