Pith. sign in

REVIEW 3 minor 42 references

Semi-discrete convex order and Laguerre tessellation fitting

T0 review · 0 major / 3 minor · reviewed 2026-06-30 · grok-4.3

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

Pith's one-line read Reconstructing a Laguerre tessellation from cell barycenters reduces to a Wasserstein projection onto discrete measures dominated in convex order by an absolutely continuous measure.

desk verdict The paper recasts Laguerre fitting as a Wasserstein projection onto convex-order dominated measures, which is a clean geometric move but rests on an AC assumption whose practical impact needs checking. read the letter →

arxiv 2606.29913 v1 pith:UHNSSFMB submitted 2026-06-29 math.OC

David P. Bourne , Thomas Gallouët , Quentin Mérigot , Andrea Natale This is my paper · ORCID
classification math.OC
keywords LaguerretessellationconvexorderWassersteinprojectionsemi-discretemeasurescellreconstructionoptimaltransportmaterialsscience
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

The paper establishes that the problem of recovering a Laguerre tessellation with prescribed cell volumes, given only the barycenters of those cells, admits a geometric reformulation. The reformulation identifies candidate discrete measures that sit below some absolutely continuous measure in the convex order. Computing the Wasserstein projection of a given discrete measure onto this set yields an approximate solution to the original reconstruction task. The same projection step also produces Laguerre fits when the input barycenters are arbitrary rather than coming from an exact tessellation. The approach is illustrated by fitting a tessellation to an electron backscatter diffraction image of steel microstructure.

What carries the argument

The set of discrete measures dominated in convex order by an absolutely continuous measure, with the Wasserstein projection onto this set serving as the approximation device for Laguerre reconstruction.

What would settle it

A concrete counter-example in which the Wasserstein projection onto the convex-order set produces cell volumes that deviate substantially from the prescribed volumes would show the approximation does not work.

Watch

Extended reading notes

Core claim

The reconstruction problem of finding a Laguerre tessellation with prescribed cell volumes from the barycenters of its cells admits a geometric interpretation as finding a discrete measure dominated in convex order by an absolutely continuous measure. The problem can therefore be solved approximately by computing the Wasserstein projection onto the set of all such discrete measures.

Load-bearing premise

The target measure must be absolutely continuous so that the convex-order domination relation is well-defined and the projection supplies a useful approximation.

Editorial extensions

If this is right

  • The exact Laguerre reconstruction problem is replaced by a tractable convex-order projection that can be computed numerically.
  • The same projection procedure yields a Laguerre tessellation fit even when the supplied barycenters do not come from any Laguerre tessellation.
  • The method directly supplies a practical tool for fitting convex partitions to experimental data such as EBSD images in materials science.

Reading between the lines

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

  • The convex-order viewpoint may extend to other semi-discrete fitting problems where cell volumes and centers must be matched simultaneously.
  • Iterative refinement around the projection could convert the approximate solution into an exact one when the data are consistent.
  • Numerical schemes for Wasserstein projection on this set could be reused as subroutines in related optimal-transport discretizations.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper studies reconstruction of a Laguerre tessellation from prescribed cell volumes and barycenters. It establishes a geometric equivalence between this problem and Wasserstein projection onto the set of discrete measures dominated in convex order by a given absolutely continuous measure, shows that the reconstruction can be solved approximately via this projection, and extends the method to fitting Laguerre tessellations to arbitrary barycenter data. A concrete application to fitting a tessellation to an EBSD image of steel microstructure is presented.

Significance. If the claimed geometric link and approximation result hold with controlled error, the work supplies a new optimal-transport route to a class of inverse problems that arise in computational geometry, imaging, and materials science. The explicit reduction to a Wasserstein projection onto a convex-order constrained set is a clean conceptual contribution; the EBSD example demonstrates immediate applicability.

minor comments (3)
  1. The abstract and introduction state that the reconstruction is solved 'approximately' by the Wasserstein projection, but the precise sense of approximation (e.g., in which metric, under what quantitative error bound) is not made explicit in the opening paragraphs; a short clarifying sentence would help readers.
  2. Notation for the convex-order domination relation and the admissible set of discrete measures should be introduced once, early, and used consistently; occasional re-definition of symbols across sections slows reading.
  3. In the materials-science application, the precise preprocessing steps that turn the EBSD image into a point cloud of barycenters and target volumes are only sketched; a short algorithmic box or pseudocode would improve reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report, so we have no specific points requiring response or revision at this stage. We will proceed with minor polishing as appropriate for the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation uses standard convex order and Wasserstein properties

full rationale

The paper's central result interprets Laguerre reconstruction as an approximate Wasserstein projection onto the convex-order dominated set. This rests on established properties of convex order (for absolutely continuous measures) and the Wasserstein metric, which are external to the paper and not defined or fitted inside it. No equations reduce a prediction to a fitted input by construction, no uniqueness theorem is imported via self-citation, and the absolute-continuity hypothesis is stated explicitly as an enabling assumption rather than smuggled in. The EBSD application is a downstream use case, not part of the derivation chain. The argument is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The work relies on background results from optimal transport and convex analysis; no new free parameters or invented entities are introduced in the abstract.

assumptions (1)
  • standard math Standard properties of convex order and Wasserstein distance between measures
    Invoked to define the feasible set and the projection operation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Semi-discrete convex order and Laguerre tessellation fitting." pith.science (2026). https://pith.science/paper/UHNSSFMB

@misc{pith2026260629913,
  author       = {Pith},
  title        = {Pith review of: Semi-discrete convex order and Laguerre tessellation fitting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UHNSSFMB}},
  note         = {Machine review of arXiv:2606.29913}
}
read the original abstract

Laguerre tessellations offer an efficient way to parameterize a large class of convex partitions of Euclidean space using only a set of points and scalar weights. For this reason, they have become popular in computational geometry, imaging and numerical analysis, both as a modeling and a discretization tool. In this paper we study the problem of reconstructing a Laguerre tessellation with prescribed cell volumes from the barycenters of its cells. We establish a geometric interpretation of this problem in terms of the set of discrete measures dominated in convex order by an absolutely continuous measure. In particular, we show that the reconstruction problem can be solved approximately by computing a Wasserstein projection onto this set. More generally, our method can also be applied to fit a Laguerre tessellation to an arbitrary set of barycenters. We give a concrete application of this in materials science, of fitting a Laguerre tessellation to an electron backscatter diffraction (EBSD) image of a steel.

Figures

Figures reproduced from arXiv: 2606.29913 by the authors.

Figure 1
Figure 1. We seek to reconstruct a Laguerre tessellation given its cells’ barycenters (i.e., their centroids) and volumes (computed with respect to a given density; here constant on a star), represented by the disks on the left. The empty dots on the right are the unknown generators of the tessellation. We refer to [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. An example of convex tessellation of R 2 which is not a (hi￾erarchical) Laguerre tessellation. The two-dimensional cells of the tes￾sellation have as boundaries the solid lines. In order for this tessellation to be Laguerre, it should be possible to recover it as the projection of a polyhedron in R 3 , but this is impossible if the dashed lines do not meet at a single point. In fact if three planes intersect in R 3 … view at source ↗
Figure 3
Figure 3. Construction of a discrete measure in convex order with a density ρ = ρ1 + ρ2 + ρ3 on R. Setting bi = bary(ρi) and vi = ρi(R), then ν3(v, B) ⪯C ρ. Lemma 3.2 (Strassen’s Theorem). Let µ, ν ∈ P1(R d ). Then µ ⪯C ν if and only if there exists a martingale coupling between µ and ν, namely, there exists a coupling θ ∈ Γ(µ, ν) such that dθ(x, y) = dθx(y)dµ(x) (in the sense of disintegration of measures) and (14) Z Rd y dθ… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Consider the Laguerre tessellation on the left, defined by the generators (y1, y2, y3) and volumes v1 = v2 = v3, where ρ is uniform on a square (shaded area). As y2 and y3 approach each other, the tessel￾lation degenerates to the configuration on the right. This limit …
Figure 5
Figure 5. Figure 5: The vector of barycenters B = (bi)i of a Laguerre tessellation with generators Y = (yi)i and volumes v = (vi)i (right figure) is the unique maximizer of X 7→ ⟨X, Y ⟩v on CN (v, ρ) (left). This proves equation (25). Finally, by direct computation and Lemma 3.1 (see also…
Figure 6
Figure 6. Figure 6: The figure on the left shows an example of a Laguerre tessel￾lation that is not irreducible. For this tessellation, the vector of barycen￾ters B = (b1, b2, b3, b4), corresponding to ρ uniform on the square and v1 = v2 = v3 = v4, lies on two distinct exposed faces of C4…
Figure 7
Figure 7. Figure 7: In order to approximate Y , the direction exposing B, we compute Y ∗ = λB − B∗ where B∗ = PCN (v,ρ)(λB) for λ > 1 close to 1. By construction, B∗ maximizes ⟨·, Y ∗ ⟩v on CN , i.e., Y ∗ exposes B∗ , and B∗ → B as λ → 1. Furthermore, we can pick Y parallel to the affine …
Figure 8
Figure 8. Figure 8: Reconstruction of a Laguerre tessellation, for t = 1250 and the subgradient descent scheme. The barycenters of the true solutions are in blue and those of the reconstruction are in red. The reconstruction obtained with the Frank-Wolfe scheme is identical (not shown) […
Figure 9
Figure 9. Figure 9: On the left: error GN (Yk) − GN (YK) for the subgradient scheme, for 1 ≤ k ≤ K, for different values of t (the curve (i) corresponds to t = 2 · 5 i−1 ), and the data in [PITH_FULL_IMAGE:figures/full_fig_p029_9.png]
Figure 10
Figure 10. Figure 10: Error of the barycenter reconstruction, ∥BK − B∥v, for dif￾ferent t at the last iteration K of (a) the subgradient descent scheme and (b) the Frank-Wolfe scheme [PITH_FULL_IMAGE:figures/full_fig_p030_10.png]
Figure 11
Figure 11. Figure 11: Reconstruction of a Laguerre tessellation, for t = 100. The barycenters of the true solution are in blue and those of the reconstruction are in red (subgradient descent, center, and Frank-Wolfe, right). acknowledges funding by the Agence Nationale de la Recherche (ANR…
Figure 12
Figure 12. Figure 12: Generators of true Laguerre tessellation in [PITH_FULL_IMAGE:figures/full_fig_p031_12.png]
Figure 13
Figure 13. Figure 13: Initial conditions for the reconstruction problem in Figures 11 and 12. The barycenters of the true solutions are in blue and those of the reconstruction are in red. [2] Aur´elien Alfonsi, Jacopo Corbetta, and Benjamin Jourdain. Sampling of one-dimensional proba￾bilit…
Figure 14
Figure 14. Figure 14: Convergence of the scheme for the data represented in Fig￾ure 11 and t = 100, shown in terms of GN (Yk) − GN (YK) for the subgra￾dient scheme (a) and JN (Bk) − JN (BK) for the Frank-Wolfe scheme (b), for 1 ≤ k ≤ K, and for the initial condition 1 (left) and 2 (right) …
Figure 15
Figure 15. Figure 15: Left: An EBSD image of steel. Right: A Laguerre tessella￾tion fitted to the areas and barycenters of the grains in the EBSD image, overlaid over the image. The barycenters of the grains are in blue, the barycenters of the fitted Laguerre cells are in red. This was com…
Figure 16
Figure 16. Figure 16: Convergence of the scheme for the data extracted from the EBSD image in [PITH_FULL_IMAGE:figures/full_fig_p033_16.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 1 canonical work pages

  1. [1]

    Kadets-type theorems for partitions of a convex body

    Arseniy Akopyan and Roman Karasev. Kadets-type theorems for partitions of a convex body. Discrete & Computational Geometry, 48:766–776, 2012. SEMI-DISCRETE CONVEX ORDER AND LAGUERRE TESSELLATIONS 31 Figure 12.Generators of true Laguerre tessellation in Figure 11 (left) and generators for reconstruction fort= 100 (subgradient descent, center, and Frank-Wol...

    2012

  2. [2]

    Aur´ elien Alfonsi, Jacopo Corbetta, and Benjamin Jourdain. Sampling of one-dimensional proba- bility measures in the convex order and computation of robust option price bounds.International Journal of Theoretical and Applied Finance, 22(03):1950002, 2019

    2019

  3. [3]

    Sampling of probability measures in the convex order by Wasserstein projection.Annales de l’Institut Henri Poincar´ e - Probabilit´ es et Statistiques, 56(3):1706–1729, 2020

    Aur´ elien Alfonsi, Jacopo Corbetta, and Benjamin Jourdain. Sampling of probability measures in the convex order by Wasserstein projection.Annales de l’Institut Henri Poincar´ e - Probabilit´ es et Statistiques, 56(3):1706–1729, 2020

    2020

  4. [4]

    A criterion for the affine equivalence of cell complexes inR d and convex polyhedra inR d+1.Discrete & Computational Geometry, 2(1):49–64, 1987

    Franz Aurenhammer. A criterion for the affine equivalence of cell complexes inR d and convex polyhedra inR d+1.Discrete & Computational Geometry, 2(1):49–64, 1987

    1987

  5. [5]

    Power diagrams: properties, algorithms and applications.SIAM Journal on Computing, 16(1):78–96, 1987

    Franz Aurenhammer. Power diagrams: properties, algorithms and applications.SIAM Journal on Computing, 16(1):78–96, 1987

    1987

  6. [6]

    Minkowski-type theorems and least- squares clustering.Algorithmica, 20(1):61–76, 1998

    Franz Aurenhammer, Friedrich Hoffmann, and Boris Aronov. Minkowski-type theorems and least- squares clustering.Algorithmica, 20(1):61–76, 1998. 32 D. P. BOURNE, T. O. GALLOU ¨ET, Q. M ´ERIGOT, AND A. NATALE Figure 14.Convergence of the scheme for the data represented in Fig- ure 11 andt= 100, shown in terms ofG N(Yk)−G N(YK) for the subgra- dient scheme (...

    1998

  7. [7]

    Bauschke and Walaa M

    Heinz H. Bauschke and Walaa M. Moursi.An Introduction to Convexity, Optimization, and Al- gorithms. SIAM, 2023

    2023

  8. [8]

    SIAM, 2017

    Amir Beck.First-order methods in optimization. SIAM, 2017

    2017

Show all 42 references
  1. [9]

    Gerald A. Beer. The Hausdorff metric and convergence in measure.Michigan Mathematical Jour- nal, 21(1):63–64, 1974

    1974

  2. [10]

    Generalized Voronoi tessellation as a model of two-dimensional cell tissue dynamics.Bulletin of Mathematical Biology, 72:1696–1731, 2010

    Martin Bock, Amit Kumar Tyagi, Jan-Ulrich Kreft, and Wolfgang Alt. Generalized Voronoi tessellation as a model of two-dimensional cell tissue dynamics.Bulletin of Mathematical Biology, 72:1696–1731, 2010

    2010

  3. [11]

    Bourne, Piet J

    David P. Bourne, Piet J. J. Kok, Steven M. Roper, and Wil D. T. Spanjer. Laguerre tessellations and polycrystalline microstructures: a fast algorithm for generating grains of given volumes. Philosophical Magazine, 100(21):2677–2707, 2020

    2020

  4. [12]

    Bourne, Mason Pearce, and Steven M

    David P. Bourne, Mason Pearce, and Steven M. Roper. Inverting Laguerre tessellations: Recov- ering tessellations from the volumes and centroids of their cells using optimal transport.ESAIM: Mathematical Modelling and Numerical Analysis, 59(5):841–871, 2025

    2025

  5. [13]

    Bourne and Steven M

    David P. Bourne and Steven M. Roper. Centroidal power diagrams, Lloyd’s algorithm, and ap- plications to optimal location problems.SIAM Journal on Numerical Analysis, 53(6):2545–2569, 2015

    2015

  6. [14]

    On optimal weighted balanced clusterings: Gravity bodies and power diagrams.SIAM Journal on Discrete Mathematics, 26(2):415–434, 2012

    Andreas Brieden and Peter Gritzmann. On optimal weighted balanced clusterings: Gravity bodies and power diagrams.SIAM Journal on Discrete Mathematics, 26(2):415–434, 2012

    2012

  7. [15]

    Constrained clustering via diagrams: A unified theory and its application to electoral district design.European Journal of Operational Research, 263(1):18–34, 2017

    Andreas Brieden, Peter Gritzmann, and Fabian Klemm. Constrained clustering via diagrams: A unified theory and its application to electoral district design.European Journal of Operational Research, 263(1):18–34, 2017

    2017

  8. [16]

    Convex optimization: Algorithms and complexity.Foundations and Trends® in Machine Learning, 8(3-4):231–357, 2015

    S´ ebastien Bubeck. Convex optimization: Algorithms and complexity.Foundations and Trends® in Machine Learning, 8(3-4):231–357, 2015

    2015

  9. [17]

    Krzysztof J. Ciosmak. Applications of Strassen’s theorem and Choquet theory to optimal transport problems, to uniformly convex functions and to uniformly smooth functions.Nonlinear Analysis, 232:113267, 2023. 34 D. P. BOURNE, T. O. GALLOU ¨ET, Q. M ´ERIGOT, AND A. NATALE

    2023

  10. [18]

    Optimal transport and the geometry ofL 1(Rd).Pro- ceedings of the American Mathematical Society, 142(10):3585–3596, 2014

    Ivar Ekeland and Walter Schachermayer. Optimal transport and the geometry ofL 1(Rd).Pro- ceedings of the American Mathematical Society, 142(10):3585–3596, 2014

    2014

  11. [19]

    An algorithm for quadratic programming.Naval research logistics quarterly, 3(1-2):95–110, 1956

    Marguerite Frank, Philip Wolfe, et al. An algorithm for quadratic programming.Naval research logistics quarterly, 3(1-2):95–110, 1956

    1956

  12. [20]

    Gallou¨ et, Andrea Natale, and Gabriele Todeschi

    Thomas O. Gallou¨ et, Andrea Natale, and Gabriele Todeschi. Metric extrapolation in the Wasser- stein space.Calculus of Variations and Partial Differential Equations, 64(5):147, 2025

    2025

  13. [21]

    On a mixture of Brenier and Strassen theorems.Proceedings of the London Mathematical Society, 120(3):434–463, 2020

    Nathael Gozlan and Nicolas Juillet. On a mixture of Brenier and Strassen theorems.Proceedings of the London Mathematical Society, 120(3):434–463, 2020

    2020

  14. [22]

    Kantorovich duality for general transport costs and applications.Journal of Functional Analysis, 273(11):3327–3405, 2017

    Nathael Gozlan, Cyril Roberto, Paul-Marie Samson, and Prasad Tetali. Kantorovich duality for general transport costs and applications.Journal of Functional Analysis, 273(11):3327–3405, 2017

    2017

  15. [23]

    Constrained clustering, diagrams, coresets, and their applications.Geometry, Analysis and Convexity, page 87, 2026

    Peter Gritzmann. Constrained clustering, diagrams, coresets, and their applications.Geometry, Analysis and Convexity, page 87, 2026

    2026

  16. [24]

    Recursively-regular subdivisions and applications.Journal of Com- putational Geometry, 7(1), 2016

    Rafel Jaume and G¨ unter Rote. Recursively-regular subdivisions and applications.Journal of Com- putational Geometry, 7(1), 2016

    2016

  17. [25]

    Statistical inference of convex order by Wasserstein projection.arXiv:2406.02840, 2024

    Jakwang Kim, Young-Heon Kim, Yuanlong Ruan, and Andrew Warren. Statistical inference of convex order by Wasserstein projection.arXiv:2406.02840, 2024

    2024

  18. [26]

    Backward and forward Wasserstein projections in stochas- tic order.Journal of Functional Analysis, 286(2):110201, 2024

    Young-Heon Kim and Yuanlong Ruan. Backward and forward Wasserstein projections in stochas- tic order.Journal of Functional Analysis, 286(2):110201, 2024

    2024

  19. [27]

    Lagrangian discretiza- tion of crowd motion and linear diffusion.SIAM Journal on Numerical Analysis, 58(4):2093–2118, 2020

    Hugo Leclerc, Quentin M´ erigot, Filippo Santambrogio, and Federico Stra. Lagrangian discretiza- tion of crowd motion and linear diffusion.SIAM Journal on Numerical Analysis, 58(4):2093–2118, 2020

  20. [28]

    Emerson Le´ on and G¨ unter M. Ziegler. Spaces of convex n-partitions. InNew Trends in Intuitive Geometry, pages 279–306. Springer, 2018

    2018

  21. [29]

    On the use of Laguerre tessellations for representations of 3d grain structures

    Allan Lyckegaard, Erik Mejdal Lauridsen, Wolfgang Ludwig, Richard Warren Fonda, and Hen- ning Friis Poulsen. On the use of Laguerre tessellations for representations of 3d grain structures. Advanced Engineering Materials, 13(3):165–170, 2011

    2011

  22. [30]

    Non-asymptotic convergence bounds for Wasserstein approximation using point clouds

    Quentin M´ erigot, Filippo Santambrogio, and Cl´ ement Sarrazin. Non-asymptotic convergence bounds for Wasserstein approximation using point clouds. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors,Proceedings of Advances in Neu- ral Infor...

    2021

  23. [31]

    Mirrors, lenses and Monge–Amp` ere equations.European Mathematical Society Magazine, (120):16–28, 2021

    Quentin M´ erigot and Boris Thibert. Mirrors, lenses and Monge–Amp` ere equations.European Mathematical Society Magazine, (120):16–28, 2021

    2021

  24. [32]

    Optimal transport: discretization and algorithms

    Quentin M´ erigot and Boris Thibert. Optimal transport: discretization and algorithms. In A. Bonito and R. H. Nochetto, editors,Geometric Partial Differential Equations - Part II, vol- ume 22 ofHandbook of Numerical Analysis, pages 133–212. North-Holland, 2021

    2021

  25. [33]

    Initialization procedures for discrete and semi-discrete optimal transport

    Jocelyn Meyron. Initialization procedures for discrete and semi-discrete optimal transport. Computer-Aided Design, 115:13–22, 2019

    2019

  26. [34]

    Reconstruction of grains in polycrystalline materials from incomplete data using Laguerre tessellations.Microscopy and Microanalysis, 25(3):743–752, 2019

    Lukas Petrich, Jakub Stanˇ ek, Mingyan Wang, Daniel Westhoff, Ludˇ ek Heller, PetrˇSittner, Carl E Krill III, Viktor Beneˇ s, and Volker Schmidt. Reconstruction of grains in polycrystalline materials from incomplete data using Laguerre tessellations.Microscopy and Microanalysi...

    2019

  27. [35]

    On the sequential convergence of Lloyd’s algorithms.Mathematics of Operations Research, 51(2):1120–1138, 2025

    Leo Portales, Elsa Cazelles, and Edouard Pauwels. On the sequential convergence of Lloyd’s algorithms.Mathematics of Operations Research, 51(2):1120–1138, 2025

    2025

  28. [36]

    Romain Quey and Lo¨ ıc Renversade. Optimal polyhedral description of 3D polycrystals: Method and application to statistical and synchrotron X-ray diffraction data.Computer Methods in Applied Mechanics and Engineering, 330:308–333, 2018

    2018

  29. [37]

    Birkh¨ auser, 2015

    Filippo Santambrogio.Optimal Transport for Applied Mathematicians. Birkh¨ auser, 2015

    2015

  30. [38]

    The existence of probability measures with given marginals.The Annals of Math- ematical Statistics, 36(2):423–439, 1965

    Volker Strassen. The existence of probability measures with given marginals.The Annals of Math- ematical Statistics, 36(2):423–439, 1965

    1965

  31. [39]

    ¨Uber exponierte punkte abgeschlossener punktmengen.Fundamenta Mathe- maticae, 24(1):139–143, 1935

    Stefan Straszewicz. ¨Uber exponierte punkte abgeschlossener punktmengen.Fundamenta Mathe- maticae, 24(1):139–143, 1935. SEMI-DISCRETE CONVEX ORDER AND LAGUERRE TESSELLATIONS 35

    1935

  32. [40]

    John F. Toland. A duality principle for non-convex optimisation and the calculus of variations. Archive for Rational Mechanics and Analysis, 71:41–61, 1979

    1979

  33. [41]

    Springer, 2009

    C´ edric Villani.Optimal transport: old and new, volume 338. Springer, 2009

    2009

  34. [42]

    Centroidal power diagrams with capacity constraints: Computation, applications, and extension

    Shi-Qing Xin, Bruno L´ evy, Zhonggui Chen, Lei Chu, Yaohui Yu, Changhe Tu, and Wenping Wang. Centroidal power diagrams with capacity constraints: Computation, applications, and extension. ACM Transactions on Graphics, 35(6):244:1–244:12, 2016. David P. Bourne, Maxwell Institut...

    2016

Pith tools

Reviewed June 30, 2026 · model on record in the stance chip above.