arXiv subjects

Ilias Ftouhi

Publications and source records attributed to Ilias Ftouhi.

11 recordsLinked to original sources

arXiv2026

Numerical exploration of the range of shape functionals using neural networks

We introduce a novel numerical framework for the exploration of Blaschke--Santaló diagrams, which are efficient tools characterizing the possible inequalities relating some given shape functionals. We introduce a parametrization of convex bodies in arbitrary dimensions using a specific invertible neural network architecture based on gauge functions, allowing an intrinsic conservation of the convexity of the sets during the shape optimization process. To achieve a uniform sampling inside the diagram, and thus a satisfying description of it, we introduce an interacting particle system that minimizes a Riesz energy functional via automatic differentiation in PyTorch. The effectiveness of the method is demonstrated on several diagrams involving both geometric and PDE-type functionals for convex bodies of $\mathbb{R}^2$ and $\mathbb{R}^3$, namely, the volume, the perimeter, the moment of inertia, the torsional rigidity, the Willmore energy, and the first two Neumann eigenvalues of the Laplacian.

math.OC↗

arXiv2026

Sensor placement via large deviations in the Eikonal equation

In this work, we address the problem of optimally placing a finite number of sensors within a given region so as to minimize the mean or maximal distance to the points of the domain. To tackle this natural geometric performance criterion, formulated in terms of distance functions, we combine tools from geometric analysis with a classical result of Varadhan, which provides an efficient approximation of the distance function via the solution of a simple elliptic PDE. The effectiveness of the proposed approach is demonstrated through illustrative numerical simulations.

math.OC↗

arXiv2025

Geometric properties of optimizers for the maximum gradient of the torsion function

Consider $J(Ω):= \|\nabla u_Ω\|_\infty/\sqrt{|Ω|} $ and $J_P(Ω):= \|\nabla u_Ω\|_\infty/P(Ω) $, where $Ω$ is a planar convex domain, $u_Ω$ is the torsion function, $P(Ω)$ is the perimeter of $Ω$ and $|Ω|$ its area. We prove that there exist planar convex domains that maximize the functionals $J$ and $J_P$, and any maximizer has a $C^1$ boundary that contains a line segment on which $|\nabla u_Ω|$ attains its maximum.

math.AP↗

arXiv2025

A reverse isoperimetric inequality for the Cheeger constant under width constraint

Henrot and Lucardesi, in Commun. Contemp. Math. (2024), conjectured that among planar convex sets with prescribed minimal width, the equilateral triangle uniquely maximizes the Cheeger constant. In this short note, we confirm this conjecture. Moreover, we establish a stability result for the inequality in terms of the Hausdorff distance.

math.OC↗

arXiv2025

The monotonicity of the Cheeger constant for parallel bodies

We prove that for every planar convex set $Ω$, the function $t\in (-r(Ω),+\infty)\longmapsto \sqrt{|Ω_t|}h(Ω_t)$ is monotonically decreasing, where $r$, $|\cdot|$ and $h$ stand for the inradius, the measure and the Cheeger constant and $(Ω_t)$ for parallel bodies of $Ω$. The result is shown to not hold when the convexity assumption is dropped. We also prove the differentiability of the map $t\longmapsto h(Ω_t)$ in any dimension and without any regularity assumption on $Ω$, obtaining an explicit formula for the derivative. Those results are then combined to obtain estimates on the contact surface of the Cheeger sets of convex bodies. Finally, potential generalizations to other functionals such as the first eigenvalue of the Dirichlet Laplacian are explored.

math.OC↗

arXiv2025

Complete systems of inequalities relating the perimeter, the area and the Cheeger constant of planar domains

The object of the paper is to find complete systems of inequalities relating the perimeter $P$, the area $|\cdot|$ and the Cheeger constant $h$ of planar sets. To do so, we study the so called Blaschke--Santaló diagram of the triplet $(P,h,|\cdot|)$ for different classes of domains: simply connected sets, convex sets and convex polygons with at most $N$ sides. We completely determine the diagram in the latter cases except for the class of convex $N$-gons when $N\ge 5$ is odd: therein, we show that the boundary of the diagram is given by the graphs of two continuous and strictly increasing functions. An explicit formula for the lower one and a numerical method to obtain the upper one is provided. At last, some applications of the results are presented.

math.OC↗

arXiv2025

The diagram $(λ_1,μ_1)$

In this paper, we are interested in the possible values taken by the pair $(λ_1(Ω), μ_1(Ω))$ the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane domain $Ω$. We prove that, without any particular assumption on the class of open sets $Ω$, the two classical inequalities (the Faber-Krahn inequality and the Weinberger inequality) provide a complete system of inequalities. Then we consider the case of convex plane domains for which we give new inequalities for the product $λ_1 μ_1$. We plot the so-called Blaschke--Santaló diagram and give some conjectures.

math.OC↗

arXiv2025

Where to place a spherical obstacle so as to maximize the first nonzero Steklov eigenvalue

We prove that among all doubly connected domains of $\mathbb{R}^n$ of the form $B_1\backslash \overline{B_2}$, where $B_1$ and $B_2$ are open balls of fixed radii such that $\overline{B_2}\subset B_1$, the first nonzero Steklov eigenvalue achieves its maximal value uniquely when the balls are concentric. Furthermore, we show that the ideas of our proof also apply to a mixed boundary conditions eigenvalue problem found in literature.

math.OC↗

arXiv2025

Improved description of Blaschke--Santaló diagrams via numerical shape optimization

We propose a method based on the combination of theoretical results on Blaschke--Santaló diagrams and numerical shape optimization techniques to obtain improved description of Blaschke--Santaló diagrams in the class of planar convex sets. To illustrate our approach, we study three relevant diagrams involving the perimeter $P$, the diameter $d$, the area $A$ and the first eigenvalue of the Laplace operator with Dirichlet boundary condition $λ_1$. The first diagram is a purely geometric one involving the triplet $(P,d,A)$ and the two other diagrams involve geometric and spectral functionals, namely $(P,λ_1,A)$ and $(d,λ_1,A)$ where a strange phenomenon of non-continuity of the extremal shapes is observed.

math.OC↗

arXiv2025

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

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.

math.OC↗

arXiv2024

Sharp inequalities involving the Cheeger constant of planar convex sets

We are interested in finding sharp bounds for the Cheeger constant $h$ via different geometrical quantities, namely the area $|\cdot|$, the perimeter $P$, the inradius $r$, the circumradius $R$, the minimal width $ω$ and the diameter $d$. We provide new sharp inequalities between these quantities for planar convex bodies and enounce new conjectures based on numerical simulations. In particular, we completely solve the Blaschke-Santaló diagrams describing all the possible inequalities involving the triplets $(P,h,r)$, $(d,h,r)$ and $(R,h,r)$ and describe some parts of the boundaries of the diagrams of the triplets $(ω,h,d)$, $(ω,h,R)$, $(ω,h,P)$, $(ω,h,|\cdot|)$, $(R,h,d)$ and $(ω,h,r)$.

math.AP↗