Pith. sign in

REVIEW 2 major objections 4 minor 38 references

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

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Optimal convex subsets in the plane have polygonal free boundaries, and the $L^p$ problem converges to Hausdorff as $p$ grows.

desk verdict Gamma-convergence part is solid, but the proof of the main regularity theorem fails for 1<p<2 due to a singular second derivative; the reader's Theorem 3 objection is mis-targeted. read the letter →

arxiv 2501.00928 v1 pith:NM76Y7MQ submitted 2025-01-01 math.OC

Zakaria Fattah , Ilias Ftouhi , Enrique Zuazua This is my paper · ORCID
classification math.OC MSC 49Q1052A2049J45
keywords shapeoptimizationconvexbodiessupportfunctionsLpapproximationGamma-convergenceHausdorffdistancepolygonalboundaryreverseisoperimetricproblem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies a shape optimization problem: given a convex container $\Omega\subset\mathbb{R}^n$, find the convex subset $\omega$ of a prescribed measure $c$ that best approximates $\Omega$ in the $L^p$ distance between support functions, $\mathcal{J}_p(\omega)=\|h_\Omega-h_\omega\|_{L^p(\mathbb{S}^{n-1})}$, for $1\le p<\infty$. The authors prove that minimizers exist in every dimension, that $\mathcal{J}_p$ $\Gamma$-converges to the Hausdorff distance $\mathcal{J}_\infty$ as $p\to\infty$, and that in the plane every optimal shape has a free boundary made of polygonal lines. The planar structural result is the paper's central claim, because it says that even smooth-looking $L^p$ approximations develop flat edges; the paper then uses that prediction to design a numerical scheme that resolves the straight segments. The contribution is the reduction of the geometric problem to an analytic one and the transfer of a known planar regularity theory to this setting.

What carries the argument

The support function $h_\omega(\theta)=\sup_{y\in\omega}\langle \theta,y\rangle$ is the central object; it encodes a convex set analytically, turns inclusion into pointwise inequality $h_\omega\le h_\Omega$, expresses area as $\frac12\int_0^{2\pi}(h^2-h'^2)\,d\theta$, and converts the geometric approximation problem into minimizing $\int_{\mathbb{S}^{n-1}}|h_\Omega-h_\omega|^p\,d\mathcal{H}^{n-1}$ subject to convexity $h_\omega''+h_\omega\ge0$. The load-bearing mechanism is the reduction captured by Theorem 3: it shows that, under the stated hypotheses, minimizing $J$ at fixed $F=x$, minimizing $J$ under $F\le x$, and minimizing $F$ at fixed $J=f(x)$ are all equivalent. That equivalence turns the measure constraint into a distance constraint and brings the planar problem into reach of a regularity theorem for optimal convex shapes, which forces the free boundary to consist of straight segments.

What would settle it

Compute $f(c)$ numerically on a fine grid for a simple container such as a triangle or disk, with a fixed $p$; if $f$ shows a strict local minimum at any interior value of $c$, the monotonicity behind Proposition 6 fails and the proof of the polygonal-boundary theorem loses its foundation.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2: in two dimensions, if $\omega^*$ solves $(P_p)$, then the free part of its boundary, $\partial\omega^*\setminus\partial\Omega$, is a union of polygonal lines; in particular, a polygonal container forces a polygonal optimizer. The proof route is Theorem 3, a general statement about two continuous shape functionals $J$ and $F$ on a compact class: under local perturbation hypotheses and a no-local-minimum condition, the value function $f(x)=\min\{J(\omega):F(\omega)=x\}$ is continuous and strictly decreasing, and the fixed-constraint problem is equivalent to the one-sided constraint problem. Applied with $F=|\cdot|$ and $J=\mathcal{J}_p$, this replaces the area constraint by a constraint on $\mathcal{J}_p$, an equivalence that lets the authors invoke a known planar regularity theory for optimal convex shapes and conclude that the free boundary is polygonal. The paper also establishes existence of minimizers and $\Gamma$-convergence to the Hausdorff problem in any dimension.

Load-bearing premise

The load-bearing premise is that the minimal approximation error $f(c)=\min\{\mathcal{J}_p(\omega):|\omega|=c\}$ is strictly decreasing in $c$; the proof of Theorem 3 derives this monotonicity from the absence of local minima, an inference that is not generally valid, and Proposition 6 uses the monotonicity to replace the measure constraint with a distance constraint.

Editorial extensions

If this is right

  • Every problem $(P_p)$ has a solution: for any convex body $\Omega\subset\mathbb{R}^n$, any $p\in[1,\infty)$, and any $c\in[0,|\Omega|]$, there is a convex $\omega\subset\Omega$ with $|\omega|=c$ attaining the infimum of $\mathcal{J}_p$.
  • As $p\to\infty$, the minimal values $\sigma_p$ converge to $\sigma_\infty$, and every Hausdorff accumulation point of $L^p$-optimal shapes solves the Hausdorff-distance problem; the $L^p$ problems are thus approximations of the limiting problem.
  • In the plane, the free part of any optimal boundary is a union of polygonal lines, and if the container is a polygon the optimizer is a polygon too.
  • At $p=1$, the problem is exactly a reverse isoperimetric problem: maximizing perimeter among convex subsets of $\Omega$ with area $c$, and every optimizer touches $\partial\Omega$ in at least two points and has straight free boundary components.
  • The numerical experiments show that the Fourier-coefficient method alone misses segments and overshoots the energy, while combining it with the discrete convexity parametrization yields lower-energy shapes with flat edges.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct numerical test of the planar theorem would run the proposed scheme on a disk container and check whether every converged optimizer contains at least one straight boundary segment; the theory predicts flat pieces for every finite $p$.
  • If the general reduction theorem is correct, the same equivalence pattern should hold for other pairs of geometric functionals, such as perimeter and area, giving a reusable template for turning constrained shape problems into one-sided ones.
  • The $\Gamma$-convergence result suggests the polygonal-boundary phenomenon is stable as $p$ grows; one could track the number and length of flat segments along a sequence of optimizers for increasing $p$.
  • In higher dimensions the natural analogue would be polyhedral free boundaries, but the paper explicitly notes the needed regularity theory is unavailable there, leaving open whether flat facets appear in $\mathbb{R}^3$ or new singularities emerge.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the shape optimization problem of finding a convex subset ω of a fixed convex container Ω, with prescribed measure, that minimizes the L^p distance between the support functions of ω and Ω. The main claims are: existence of minimizers and Γ-convergence to the Hausdorff-distance problem as p→∞ (Theorem 1); in the plane, the free part of any optimal boundary is a union of polygonal lines (Theorem 2); and an abstract equivalence theorem (Theorem 3) that is used to replace the measure constraint by a distance constraint in the proof of Theorem 2. The paper also proposes a numerical scheme combining Fourier parametrization with a convexity-preserving discretization.

Significance. The Γ-convergence result and the numerical method are useful contributions, and the polygonal-regularity statement is natural and would be significant if proven. However, the central structural result (Theorem 2) rests on an abstract theorem (Theorem 3) that is false as stated, and on a second-order estimate that fails for 1<p<2. These are load-bearing gaps, not presentation issues, so the paper's main theoretical claims are not established.

major comments (2)
  1. [Section 3.3, Proposition 11] Theorem 3 is false as stated. The proof of Proposition 11 only shows that f has no local minimum in the interior of I, and then concludes that f is strictly decreasing. This inference is invalid: a continuous function on an interval with no interior local minima may have local maxima or be increasing. A concrete counterexample satisfying hypotheses (A)-(D) is C = {[0,t] : t∈[0,1]} with the Hausdorff distance, F([0,t])=t, J([0,t])=t, and Ψ_{[0,t]}(x)=[0,x]. Then I=[0,1], all hypotheses hold, and f(t)=t is strictly increasing, contradicting the asserted conclusion. Consequently, Proposition 6, which relies on Theorem 3, is not proven, and the derivation of the equivalent formulation used in the proof of Theorem 2 is unsupported.
  2. [Section 3.2, proof of Theorem 7] The estimate ||m''(h)(v,v)||_{L^2} ≤ β||v||^2_{L^2} is false for 1<p<2. With m(h)=∫(hΩ-h)^p dθ, the second derivative is m''(h)(v,v)=p(p-1)∫(hΩ-h)^{p-2}v^2 dθ. When p<2, the weight (hΩ-h)^{p-2} is unbounded near the contact set {h=hΩ}. For a sequence v_n supported on an interval of length 1/n near a contact point with ||v_n||_{L^2}=1, the integral grows like n^{2-p}, so the L^2 bound fails. Since Theorem 2 is claimed for all p∈[1,∞), the application of [20, Theorem 2.9] is not justified for 1<p<2. The separate treatment of p=1 in Proposition 13 invokes Theorem 2 and therefore inherits this gap.
minor comments (4)
  1. [Section 3.2, equation (8)] The constraint in problem (8) is written as ∫(hΩ-h)^p dθ = f(c), but since J_p is defined as the p-th root of the integral, the correct constraint should involve f(c)^p (or a renamed constant). This is a notational inconsistency that should be fixed.
  2. [Section 3.3, proof of Proposition 11] In the inferior-limit part of the continuity proof, the line 'F(Ω∗)=x' should read 'F(Ω∗)=x0', since the limit point is x0.
  3. [Section 4, Conjecture 1] The angle notation '[CAB ≥ [CBA' is garbled; it should be written as ∠CAB ≥ ∠CBA.
  4. [Section 1, introduction] There is a typo 'on an other note' which should be 'on another note'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning found; the main weakness is an incorrect L2 bound in the proof of Theorem 2 for 1<p<2, which is a correctness gap rather than a circular reduction.

full rationale

The derivation chain is not circular. Theorem 1 is a standard compactness plus Gamma-convergence argument, self-contained. Theorem 2 relies on the external regularity theorem [20, Theorem 2.9] after Proposition 6 establishes, via the internally proved Theorem 3, an equivalence between (P_p) and a volume-minimization problem with fixed p-distance. Theorem 3 is proved in the paper (Propositions 9-12); although the monotonicity step in Proposition 11 is written too quickly (absence of local minima does not by itself imply strict monotonicity), that is a logical gap, not a circular definition or fitted-input disguised as a prediction. The only self-citations are to [17] (Ftouhi-Zuazua) for the p=infinity Hausdorff problem, and these are peripheral: the p=infinity equivalence is also recoverable from the paper's own Proposition 6, and the planar regularity proof uses [20], which is external and parameter-free. The serious mathematical issue flagged by the skeptic is in the proof of Theorem 2 for 1<p<2: the asserted bound ||m''(h)(v,v)|| <= beta ||v||^2_{L2} is false when (h_Omega-h)^{p-2} is unbounded near the contact set. This affects the correctness of Theorem 2, but it is not an instance of the paper's output being equivalent to its input by construction. No parameter is fitted and then renamed as a prediction, and no uniqueness conclusion is imported solely from the authors' own prior work. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper relies on standard convex geometry results, an imported regularity theorem, and a false logical step. No new entities or fitted parameters are introduced. The main burden is the false implication in Theorem 3, which propagates to Proposition 6 and Theorem 2.

assumptions (7)
  • standard math Blaschke selection theorem provides compactness of the class of convex subsets of a bounded container
    Used to verify Hypothesis (A) of Theorem 3 and to prove existence in Theorem 1.
  • standard math Continuity of area and J_p with respect to Hausdorff distance
    Verifies Hypothesis (B) of Theorem 3; J_p continuity is proved via a Lipschitz bound by d_H.
  • domain assumption For every set in C there is a continuous path Psi with F(Psi(t))=t (Hypothesis (C))
    For the application, this map is constructed using inner parallel sets and convex combinations; the abstract theorem assumes this generally.
  • domain assumption The perturbation (1-epsilon)omega+epsilon Omega strictly decreases J_p, giving Hypothesis (D)
    Shown in Proposition 6 for the specific functionals, and used to rule out local minima.
  • ad hoc to paper The implication 'no local minima implies strictly decreasing' for continuous functions on an interval
    This step in Proposition 11 is false; it is the invalid inference that breaks Theorem 3.
  • domain assumption Regularity theorem [20, Theorem 2.9] for optimal convex shapes
    Imported from Lamboley, Novruzi, and Pierre; used to deduce polygonal free boundaries in Theorem 2.
  • domain assumption Uniform convexity condition h_Omega'' + h_Omega >= d0 in Proposition 15
    Assumed so that h_Omega - d is a support function; restricts the class of containers for which inner parallel sets are optimal.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimal $L^p$-approximation of convex sets by convex subsets." pith.science (2026). https://pith.science/paper/NM76Y7MQ

@misc{pith2026250100928,
  author       = {Pith},
  title        = {Pith review of: Optimal $L^p$-approximation of convex sets by convex subsets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NM76Y7MQ}},
  note         = {Machine review of arXiv:2501.00928}
}
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.

Figures

Figures reproduced from arXiv: 2501.00928 by the authors.

Figure 1
Figure 1. The support function of the convex Ω. The support functions of planar convex sets have some interesting properties [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Parallel chord movements increase the perimeter while pre￾serving the area. At last, since the free part of the boundary of the optimal set ω ∗ is a line, then ∂Ω∩ω ∗ contains at least two different points. □ Remark 14. It is straightforward that if the container Ω is a polygon then the optimal solution ω ∗ is also polygonal. We note that Problem (10) has recently been completely solved by B. Bogosel when the contai… view at source ↗
Figure 3
Figure 3. The triangle MAB is conjectured to be the solution of the reverse isoperimetric problem when the contained Ω is the triangle ABC. 4.2. The case p = +∞. The problem (P∞) can be written as min{d H(ω, Ω) | ω ⊂ Ω is convex and |ω|= c}, which is, as stated in [17, Theorem 1], equivalent to the problem (11) min{|ω| | ω is convex and hΩ − dc ≤ hω ≤ hΩ}, where dc is a constant depending on c. We are able to characterize the… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: First example for the history of convergence of the two meth￾ods (with a zoom in the figure in right). Method 1 Method 2 Energy = 1.185 Energy = 1.053 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: Second example for the history of convergence of the two methods (with a zoom in the figure in right) [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: Varadhan’s result for the approximation of the distance func￾tion [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 37 canonical work pages

  1. [1]

    E. G. Alverado, B. Krishnamoorthy, and K. R. Vixie. The maximum distance problem and mini- mum spanning trees. International Journal of Analysis and Applications , 19(5):633–659, 2021

    work page 2021

  2. [2]

    P. R. S. Antunes and B. Bogosel. Parametric shape optimization using the support function. Comput. Optim. Appl. , 82(1):107–138, 2022

    work page 2022

  3. [3]

    Bianchini and A

    C. Bianchini and A. Henrot. Optimal sets for a class of minimization problems with convex constraints. J. Convex Anal. , 19(3):725–758, 2012

    work page 2012

  4. [4]

    B. Bogosel. Numerical shape optimization among convex sets. Appl. Math. Optim. , 87(1):Paper No. 1, 31, 2023

    work page 2023

  5. [5]

    B. Bogosel. A reverse isoperimetric inequality for convex shapes with inclusion constraint.J. Math. Anal. Appl., 540(2):14, 2024. Id/No 128636

    work page 2024

  6. [6]

    A. Braides. Local minimization, variational evolution and Γ-convergence, volume 2094 of Lect. Notes Math. Cham: Springer, 2014

    work page 2014

  7. [7]

    Buttazzo, E

    G. Buttazzo, E. Oudet, and E. Stepanov. Optimal transportation problems with free Dirichlet regions. In Variational methods for discontinuous structures. International workshop, Villa Erba (Cernobbio), Italy, July 4–6, 2001 , pages 41–65. Basel: Birkh¨ auser, 2002

    work page 2001

  8. [8]

    Buttazzo and E

    G. Buttazzo and E. Stepanov. Optimal transportation networks as free Dirichlet for the Monge- Kantorovich problem. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) , 2(4):631–678, 2003

    work page 2003

Show all 38 references
  1. [9]

    Cherkashin and Y

    D. Cherkashin and Y. Teplitskaya. On the horseshoe conjecture for maximal distance minimizers. ESAIM, Control Optim. Calc. Var. , 24(3):1015–1041, 2018

    2018

  2. [10]

    Crane, C

    K. Crane, C. Weischedel, and M. Wardetzky. Geodesics in heat: A new approach to computing distance based on heat flow. ACM transactions on graphics , 32:826–835, 2013

    2013

  3. [11]

    Croce, Z

    G. Croce, Z. Fattah, and G. Pisante. A reverse isoperimetric inequality for planar ( α, β)-convex bodies. ESAIM, Control Optim. Calc. Var. , 28:18, 2022. Id/No 62

    2022

  4. [12]

    Drach and K

    K. Drach and K. Tatarko. Reverse isoperimetric problems under curvature constraints. Preprint, arXiv:2303.02294, 2023

    2023 arXiv

  5. [13]

    J. Favard. Probl` emes d’extremums relatifs aux courbes convexes. I. Ann. Sci. ´Ec. Norm. Sup´ er. (3), 46:345–369, 1929

    1929

  6. [14]

    A. Florian. On a metric for the class of compact convex sets. Geom. Dedicata, 30(1):69–80, 1989

    1989

  7. [15]

    I. Ftouhi. Where to place a spherical obstacle so as to maximize the first nonzero Steklov eigen- value. ESAIM Control Optim. Calc. Var. , 28:Paper No. 6, 21, 2022

    2022

  8. [16]

    Ftouhi and J

    I. Ftouhi and J. Lamboley. Blaschke-Santal´ o diagram for volume, perimeter, and first Dirichlet eigenvalue. SIAM J. Math. Anal. , 53(2):1670–1710, 2021

    2021

  9. [17]

    Ftouhi and E

    I. Ftouhi and E. Zuazua. Optimal design of sensors via geometric criteria. J. Geom. Anal. , 33(8):Paper No. 253, 29, 2023

    2023

  10. [18]

    E. M. II Harrell and A. Henrot. On the maximization of a class of functionals on convex regions, and the characterization of the farthest convex set. Mathematika, 56(2):245–265, 2010. 26 ZAKARIA F ATTAH, ILIAS FTOUHI, AND ENRIQUE ZUAZUA

    2010

  11. [19]

    Howard and A

    R. Howard and A. Treibergs. A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature. Rocky Mt. J. Math. , 25(2):635–684, 1995

    1995

  12. [20]

    Lamboley, A

    J. Lamboley, A. Novruzi, and M. Pierre. Regularity and singularities of optimal convex shapes in the plane. Arch. Ration. Mech. Anal. , 205(1):311–343, 2012

    2012

  13. [21]

    Lamboley, A

    J. Lamboley, A. Novruzi, and M. Pierre. Estimates of first and second order shape derivatives in nonsmooth multidimensional domains and applications. J. Funct. Anal., 270(7):2616–2652, 2016

    2016

  14. [22]

    Lemenant

    A. Lemenant. A presentation of the average distance minimizing problem. J. Math. Sci., New York, 181(6):820–836, 2012

    2012

  15. [23]

    Lemenant and E

    A. Lemenant and E. Mainini. On convex sets that minimize the average distance. ESAIM, Control Optim. Calc. Var. , 18(4):1049–1072, 2012

    2012

  16. [24]

    Mazari, G

    I. Mazari, G. Nadin, and Y. Privat. Optimal location of resources maximizing the total population size in logistic models. J. Math. Pures Appl. (9) , 134:1–35, 2020

    2020

  17. [25]

    Mazzoleni, B

    D. Mazzoleni, B. Pellacci, and G. Verzini. Asymptotic spherical shapes in some spectral optimiza- tion problems. J. Math. Pures Appl. (9) , 135:256–283, 2020

    2020

  18. [26]

    Modica and S

    L. Modica and S. Mortola. Un esempio di Γ −-convergenza. Boll. Unione Mat. Ital., V. Ser., B , 14:285–299, 1977

    1977

  19. [27]

    Privat, E

    Y. Privat, E. Tr´ elat, and E. Zuazua. Optimal location of controllers for the one-dimensional wave equation. Ann. Inst. Henri Poincar´ e, Anal. Non Lin´ eaire, 30(6):1097–1126, 2013

    2013

  20. [28]

    Privat, E

    Y. Privat, E. Tr´ elat, and E. Zuazua. Optimal shape and location of sensors for parabolic equations with random initial data. Arch. Ration. Mech. Anal. , 216(3):921–981, 2015

    2015

  21. [29]

    Privat, E

    Y. Privat, E. Tr´ elat, and E. Zuazua. Actuator design for parabolic distributed parameter systems with the moment method. SIAM J. Control Optim. , 55(2):1128–1152, 2017

    2017

  22. [30]

    Roques and F

    L. Roques and F. Hamel. Mathematical analysis of the optimal habitat configurations for species persistence. Math. Biosci. , 210(1):34–59, 2007

    2007

  23. [31]

    Schneider

    R. Schneider. Convex Bodies: The Brunn-Minkowski Theory . Cambridge University Press, 2nd expanded edition edition, 2013

    2013

  24. [32]

    Teplitskaya

    Y. Teplitskaya. Regularity of maximum distance minimizers. J. Math. Sci., New York, 232(2):164– 169, 2018

    2018

  25. [33]

    S. R. S. Varadhan. On the behavior of the fundamental solution of the heat equation with variable coefficients. Commun. Pure Appl. Math. , 20:431–455, 1967

    1967

  26. [34]

    R. A. Vitale. Lp metrics for compact convex sets. J. Approx. Theory , 45:280–287, 1985

    1985

  27. [35]

    Wang and E

    G. Wang and E. Zuazua. On the equivalence of minimal time and minimal norm controls for internally controlled heat equations. SIAM J. Control Optim. , 50(5):2938–2958, 2012. (Zakaria F attah) Mathematics and Computer Science Department, ENSAM of Meknes, Univer- sity of Moulay ...

    2012

  28. [36]

    King F ahd university of Petroleum and Minerals, Department of Mathematics, 31261 Dhahran, Saudi Arabia. Email address: ilias.ftouhi@fau.de (Enrique Zuazua) [1] Chair for Dynamics, Control, Machine Learning and Numerics, Alexan- der von Humboldt-Professorship, Department of Ma...

  29. [37]

    de las Universidades, 24, 48007 Bilbao, Basque Country, Spain,

    Chair of Computational Mathematics, Fundaci ´on Deusto, A v. de las Universidades, 24, 48007 Bilbao, Basque Country, Spain,

  30. [38]

    Email address: enrique.zuazua@fau.de

    Departamento de Matem´aticas, Universidad Aut ´onoma de Madrid, 28049 Madrid, Spain. Email address: enrique.zuazua@fau.de

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.