{"id":1419,"date":"2025-08-22T16:09:53","date_gmt":"2025-08-22T16:09:53","guid":{"rendered":"https:\/\/summergeometry.org\/sgi2025\/?p=1419"},"modified":"2025-08-25T00:44:32","modified_gmt":"2025-08-25T00:44:32","slug":"couplings-not-courtiers-the-convex-way-to-feed-a-kingdom","status":"publish","type":"post","link":"https:\/\/summergeometry.org\/sgi2025\/couplings-not-courtiers-the-convex-way-to-feed-a-kingdom\/","title":{"rendered":"Couplings, Not Courtiers: The Convex Way to Feed a Kingdom"},"content":{"rendered":"\n

\ud83c\udfad A Tale of Two Transports<\/em>: A Comic Drama in Three Acts<\/strong> \ud83c\udfad<\/p>\n\n\n\n

Dramatis Personae<\/strong><\/h6>\n\n\n\n
    \n
  • King Louis XVI<\/strong> \u2014 monarch of France; dislikes chaos and prefers order.<\/li>\n\n\n\n
  • Peasant<\/strong> \u2014 voice of everyday inefficiency.<\/li>\n\n\n\n
  • Gaspard Monge <\/strong>\u2014 French mathematician; formulates transport as a deterministic map with elegant clarity.<\/li>\n\n\n\n
  • Party Official<\/strong> \u2014 a bureaucrat, loves words like \u201cplan.\u201d<\/li>\n\n\n\n
  • Leonid Kantorovich<\/strong> \u2014 Soviet mathematician; introduces the convex relaxation that makes transport broadly solvable.<\/li>\n\n\n\n
  • Yann Brenier <\/strong>\u2014 mathematician; shows when the optimal plan is in fact a gradient map of a convex potential.<\/li>\n<\/ul>\n\n\n\n

    Act I \u2014 A Village in France, 1780<\/strong><\/h2>\n\n\n\n

    Scene: Grey dawn. Carts piled with wheat creak by; a boy staggers under chiming milk pails; two men shoulder grapes toward a wine press heroically far from grapes. Chickens run quality assurance on everyone\u2019s ankles. A Peasant collapses onto a sack. Enter Louis XVI<\/strong>, in a plain cloak, incognito, attempting \u201cpeasant walk.\u201d<\/em><\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    \ud83d\udc51 Louis XVI<\/strong> <\/em>(softly, to himself):<\/strong> Ah, the countryside\u2014so tranquil. (A wheelbarrow nearly flattened<\/em> him.)<\/em> \u2026 Enchanting.<\/p>\n\n\n\n

    \u201cPourquoi<\/em>,\u201d the Louis XVI <\/em>asked a peasant who had stopped to breathe\u2014and to reconsider his life choices\u2014\u201cis everyone running like headless hens? Why must you drag wheat halfway across France?\u201d<\/p>\n\n\n\n

    \ud83d\udc68\u200d\ud83c\udf3ePeasant (gasping):<\/strong> New regulations\u2014( he does not recognize him)<\/em>\u2014each farm must send all<\/strong> its wheat to the Rouen mill. Jean\u2019s milk goes all<\/strong> beyond the river. Pierre\u2019s grapes go all<\/strong> to that press on Mount Ridicule. Each farm is assigned to a factory regardless of whether it\u2019s nearby or not. It is the system<\/strong>.<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    \ud83d\udc51 Louis XVI<\/strong> <\/em>(deadpan):<\/strong> System.<\/em> Like umbrellas made of lace.<\/p>\n\n\n\n

    \ud83d\udc68\u200d\ud83c\udf3e Peasant:<\/strong> They call it \u201cRational Logistics.\u201d The rational part is the paperwork. The logistics we do on foot.<\/p>\n\n\n\n

    \ud83d\udc51 Louis XVI<\/strong> <\/em><\/strong>:<\/strong> And why not send wheat to the nearest mill?<\/p>\n\n\n\n

    \ud83d\udc68\u200d\ud83c\udf3ePeasant:<\/strong> Nearest? (laughs, a little unwell)<\/em> The Committee for Extended Journeys of Grain has determined \u201cnearest\u201d is bourgeois. We ship by destiny<\/strong>, not distance.<\/p>\n\n\n\n

    \ud83d\udc68\u200d\ud83c\udf3ePeasant (rising, grimly cheerful):<\/strong> Must be off. My route goes past Rouen, then back past here, then past my grave, then the mill.<\/p>\n\n\n\n

    \ud83d\udc51 Louis XVI<\/strong> <\/em>:<\/strong> Bon voyage. (beat)<\/em> And bon sens\u2026 someday.<\/p>\n\n\n\n

    The Peasant trundles away, pursued by chickens. The King stares at the chaos like a man realizing his kingdom is a long proof with a missing lemma. He turns on his heel.<\/em> Exit Louis XVI.<\/strong><\/p>\n\n\n\n

    At the Royal Palace<\/em><\/strong><\/p>\n\n\n\n

    Scene: A long table buried in maps. Brass compasses glint. Officials murmur in hexagons, puzzled. Enter Monge<\/strong>, ink on cuffs; Louis XVI<\/strong> presides.<\/em><\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    \ud83d\udc51 Louis XVI<\/strong> <\/em>(curt):<\/strong> The villages in France are ruled by chaos. Help us fix it.<\/p>\n\n\n\n

    Monge (with the unstoppable serenity of a man who believes the world can be improved by equations.):<\/strong> The problem is assigning yields (wheat, milk, grapes) from farms in France to the appropriate factories (flour mills, dairies, wine presses) at minimal total transport cost. That is optimal transport <\/em>similar to the one with mines and factories that I was working on.<\/em> Let me show you how to solve it with wheat and mills.<\/p>\n\n\n\n

    Monge (to Louis XVI<\/strong> <\/em><\/strong>):<\/strong> Let us treat the country map as two mathematical worlds: \\(X\\) for farms and \\(Y\\) for mills. Plainly, \\(Y\\) and \\(X\\) are just set of points on your map (farm and mill locations), and any blob you can draw around those points is a region in which we might count wheat or capacity.<\/p>\n\n\n\n

    Precisely, take \\(X\\) and \\(Y\\) \\( \\in \\mathbb{R}^2 \\) (the plane of the map) and equip them with their restricted Borel \\(\\sigma\\) -algebras:
    \\[
    \\mathcal{B}_X:=\\{U\\cap X:\\ U\\text{ open in }\\mathbb{R}^2\\},\\qquad
    \\mathcal{B}_Y:=\\{V\\cap Y:\\ V\\text{ open in }\\mathbb{R}^2\\}.
    \\]
    And there you have two measurable spaces: \\((X,\\mathcal{B}_X)\\) and \\((Y,\\mathcal{B}_Y)\\)<\/p>\n\n\n\n

    These \\(\\sigma\\)-algebras specify which regions count as the measurable regions <\/em>on which amounts can be tallied; the individual farm\/mill points are measurable singletons and will serve as atoms<\/em>.<\/p>\n\n\n\n

    Atomic data and finite Borel measures.<\/strong>
    Given farms at points \\(x_i\\in X\\) with supplies \\(s_i\\ge0\\) and mills at points \\(y_j\\in Y\\) with capacities
    \\(d_j\\ge0\\), define finite Borel measures (here \\(\\delta_x\\) is the Dirac unit mass at \\(x\\)):
    \\[
    \\tilde\\mu=\\sum_i s_i \\, \\delta_{x_i}\\\\ \\text{on}(X,\\mathcal{B}_X),<\/em> \\qquad \\tilde\\nu=\\sum_j d_j \\, \\delta_{y_j} \\ \\\\text{on }(Y,\\mathcal{B}_Y).
    \\]
    Assume mass balance<\/em> \\(\\tilde\\mu(X)=\\tilde\\nu(Y)=:S\\in(0,\\infty)\\).<\/p>\n\n\n\n

    Normalize for clean bookkeeping.<\/strong>
    For cleaner formulas set
    \\[
    \\mu:=\\tilde\\mu\/S=\\sum_i \\frac{s_i}{S}\\,\\delta_{x_i},\\qquad
    \\nu:=\\tilde\\nu\/S=\\sum_j \\frac{d_j}{S}\\,\\delta_{y_j},
    \\]
    so \\(\\mu(X)=\\nu(Y)=1\\).<\/p>\n\n\n\n

    Interpretation:<\/strong> \\(\\mu(A)\\) is the fraction of national wheat in \\(A\\subset X\\), and \\(\\nu(B)\\) the fraction of national milling
    capacity in \\(B\\subset Y\\). (Normalization rescales costs by \\(S\\).)<\/p>\n\n\n\n

    In this problem, we can work with a purely discrete model;
    We only care about the finitely many sites, thus we may take
    \\[
    X={x_1,\\dots,x_m},\\ \\mathcal{B}_X=2^X,\\qquad
    Y={y_1,\\dots,y_n},\\ \\mathcal{B}_Y=2^Y,
    \\]
    the power-set (\\sigma\\)-algebras.<\/p>\n\n\n\n

    Monge (drawing crisp arrows):<\/strong> <\/strong>Next, we declare <\/strong>a measurable, lower semicontinuous cost rule \\(c:X\\times Y\\to[0,\\infty]\\); typical choices are \\(c(x,y)=|x-y|\\) or \\(c(x,y)=\\frac{1}{2} \\|x-y\\|^2\\). The per\u2013unit bill to haul from \\(x\\) to \\(y\\) is \\(c(x,y)\\).<\/p>\n\n\n\n

    Monge continues ..<\/strong>
    Then my principle is: one farm, one address via a map <\/em>\\(T:X\\to Y\\)<\/strong>
    Each farm sends all <\/em>its wheat to exactly one mill. Mathematically, this means we find a measurable map \\(T:X\\to Y\\) with the pushforward constraint \\(T_{\\#}\\mu=\\nu\\) minimizing
    \\[
    \\ \\min_{T_{\\#}\\mu=\\nu}\\ \\int_X c\\big(x,T(x)\\big)\\,d\\mu(x)\\ \\quad\\text{(no splitting of mass).}
    \\]
    Probabilistic reading: if \\(X\\sim\\mu\\) and \\(Y:=T(X)\\), then \\(Y\\sim\\nu\\); the induced coupling is concentrated on the graph \\({(x,T(x))}\\).<\/p>\n\n\n\n

    \ud83d\udc51 Louis XVI<\/strong> <\/em><\/strong> (applauding):<\/strong> One clean arrow from each farm to one mill. Very elegant. Very royal. Also, occasionally disastrous\u2014like sending three tons from one farm to a single mill that can only swallow one.<\/p>\n\n\n\n

    Monge (unruffled): <\/strong>My constraint really is what it says: a map does not split atoms of mass. Elegance has limits.<\/p>\n\n\n\n

    Act II \u2014 Moscow, 1939<\/strong><\/h2>\n\n\n\n

    Scene: A Soviet planning office. Maps of farms and bread factories wallpaper the room. A portrait watches. A heavy desk. A heavier silence. Party Official<\/strong> sits; a clerk guards a terrifying spreadsheet. Enter Kantorovich<\/strong>, notebook in hand.<\/em><\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    \u262d Party Official (to Kantorovich):<\/strong> Wheat moves. Bread appears. Costs explode. Could you explain how the first two can occur without the third?<\/p>\n\n\n\n

    \ud83e\uddee Kantorovich:<\/strong>
    Hmm, it is the Monge\u2019s French approach to optimal transport. This method allows only one-to-one assignments \u2014 a farm to a single factory. It is beautiful\u2014like a court etiquette manual. It also breaks the budget<\/strong> when capacities don\u2019t line up.<\/p>\n\n\n\n

    \u262d Official:<\/strong> Examples?<\/p>\n\n\n\n

    \ud83e\uddee Kantorovich:<\/strong> Farm A has 2 tons; Mill 1 and Mill 2 each need 1 ton.
    Monge\u2019s rule can\u2019t split<\/strong> A\u2014so either we overflow a mill or we ship the extra far away at great cost. It is time we shall stop assigning by etiquette and start assigning by fractions<\/strong>. One farm\u2019s harvest may be shared among Leningrad, Moscow, and Kiev (several mills). No drama, no lace.

    \u262d Official (suspicious):<\/strong>
    Shared? You propose to slice the wheat like bourgeois cake?<\/p>\n\n\n\n

    \ud83e\uddee Kantorovich (calmly):<\/strong>
    Not cake\u2014convexity.<\/em><\/p>\n\n\n\n

    \u262d Official:<\/strong> Could you explain your method. How do we schedule wheat to mills?<\/p>\n\n\n\n

    \ud83e\uddee Kantorovich<\/strong>: We keep the same data: farms live in \\(X\\) with supply measure \\(\\mu\\), mills live in \\(Y\\) with capacity measure \\(\\nu\\), and \\(c(x,y)\\) is the per\u2013unit haul cost. Normalize so \\(\\mu(X)=\\nu(Y)=1).\\)<\/p>\n\n\n\n

    We will not do a single address per farm, but a plan \\(\\pi\\): it tells what fraction of wheat goes from each farm \\(x\\) to each mill \\(y\\).
    Mathematically, \\(\\pi\\) is a measure on \\(X\\times Y\\) with the correct row and column totals (marginals):
    \\[\\Pi(\\mu,\\nu):=\\Big\\{\\pi\\text{ on }X\\times Y:\\ \\pi(A\\times Y)=\\mu(A),\\ \\ \\pi(X\\times B)=\\nu(B)\\ \\ \\forall A\\in\\mathcal B_X,\\ B\\in\\mathcal B_Y\\Big\\}.\\]<\/p>\n\n\n\n

    Spreadsheet intuition<\/strong>: rows = farms, columns = mills; the entry \\(\\pi(x,y)\\) is the fraction shipped \\(x\\to y\\).
    Row sums match \\(\\mu\\) (every farm ships all its wheat); column sums match \\(\\nu\\) (every mill is filled exactly).<\/p>\n\n\n\n

    \u262d Official<\/strong>: And we choose \\(\\pi\\) how?

    \ud83e\uddee Kantorovich<\/strong>: By minimizing the national bill (the expected cost under \\(\\pi)\\):

    \\[
    \\min_{\\pi\\in\\Pi(\\mu,\\nu)} \\ \\int_{X\\times Y} c(x,y)\\,d\\pi(x,y)\\ \\qquad \\text{(mass may split).}\\]

    If the country\u2019s total wheat is \\(S\\) tons (before normalization), the physical cost is \\(S!\\int c\\,d\\pi\\).<\/p>\n\n\n\n

    \u262d Official<\/strong>: Why is this reliable?<\/p>\n\n\n\n

    \ud83e\uddee Kantorovich<\/strong>: Because the feasible set \\(\\Pi(\\mu,\\nu)\\) is convex (mixtures of schedules are schedules) and the objective is linear in \\(\\pi\\). On reasonable spaces \\(Polish (X,Y)\\) (e.g.\\ subsets of \\(\\mathbb R^2\\)) with a lower semicontinuous cost \\(c\\) (no sudden discounts), an optimal plan exists. Always.<\/p>\n\n\n\n

    \u262d Official<\/strong> (nodding): Fractions in the boxes, rows = supply, columns = demand, minimize the sum. No lace.<\/p>\n\n\n\n

    \ud83e\uddee Kantorovich<\/strong> (smiles): Exactly. More bread, fewer marathons.<\/p>\n\n\n\n

    \u262d Official<\/strong>: And how do we know which roads actually carry wheat? I dislike paying for decorations.<\/p>\n\n\n\n

    \ud83e\uddee Kantorovich<\/strong>: We use shadow prices.<\/em> Assign a number to each farm and each mill:
    \\[
    \\varphi:X\\to\\mathbb R\\quad(\\text{farms}),\\qquad \\psi:Y\\to\\mathbb R\\quad(\\text{mills}),
    \\]
    with the global no-underpricing rule
    \\[
    \\varphi(x)+\\psi(y)\\le c(x,y)\\quad\\text{for all }(x,y).
    \\]
    Then we maximize the total value
    \\[
    \\sup_{\\varphi,\\psi}\\ \\Big(\\int_X \\varphi\\,d\\mu+\\int_Y \\psi\\,d\\nu\\Big)\\ \\text{ subject to }\\ \\varphi+\\psi\\le c\\ .
    \\]
    Strong duality holds: this supremum equals the minimal transport cost. At the optimal schedule \\(\\pi^\\ast\\),
    \\[
    \\varphi^\\ast(x)+\\psi^\\ast(y)=c(x,y)\\quad\\text{for }\\pi^\\ast\\text{-a.e. }(x,y),
    \\]
    so wheat flows only on tight routes. Overpriced routes carry no flow.<\/p>\n\n\n\n

    \u262d Official<\/strong>: Good. Prices that turn lights on where grain should move. When do we get back to one arrow per farm?<\/p>\n\n\n\n

    \ud83e\uddee Kantorovich<\/strong>: When nature is kind. If supply isn\u2019t all concentrated at points and the cost rewards straight, nearby routes, the best schedule often behaves like a single address per source\u2014effectively a map. But that conclusion belongs to future mathematics, Comrade; today we use the plan.<\/p>\n\n\n\n

    \u262d Official<\/strong>: And what do the clerks actually compute while I frown at the portrait?<\/p>\n\n\n\n

    \ud83e\uddee Kantorovich<\/strong>: A linear program. With farms \\(i\\), mills \\(j\\), shipments \\(x_{ij}\\ge0\\), costs \\(c_{ij}=c(x_i,y_j)\\):
    \\[
    \\begin{aligned}
    \\min_{x_{ij}\\ge0}\\quad & \\sum_{i,j} c_{ij}\\,x_{ij} \\
    \\text{s.t.}\\quad & \\sum_j x_{ij}=s_i\\quad(\\text{ship all supply}),\\qquad
    \\sum_i x_{ij}=d_j\\quad(\\text{fill all mills}).
    \\end{aligned}
    \\]
    Normalize by \\(S=\\sum_i s_i): (\\pi_{ij}:=x_{ij}\/S\\) is a discrete coupling in \\(\\Pi(\\mu,\\nu)\\) and
    \\(\\sum c_{ij}x_{ij}=S\\int c\\,d\\pi\\).
    They can solve it by transportation algorithms; your signature appears only at the end.<\/p>\n\n\n\n

    \u262d Official<\/strong>(closing the file): Prices to light the roads, plans to split the loads, and arrows when the heavens permit. Acceptable.<\/p>\n\n\n\n

    \ud83e\uddee Kantorovich<\/strong>(dry): Monge when you can, coupling when you must. Either way, warm bread\u2014without marathons.<\/p>\n\n\n\n

    The clerk exhales. The portrait approves. The spreadsheet looks slightly less terrifying.<\/em><\/p>\n\n\n\n

    Act III \u2014 Paris, 1991 <\/h2>\n\n\n\n

    Scene<\/em>: Seminar room; chalk dust in sunbeams. A blackboard already half-conquered. Enter Yann Brenier<\/strong>, chalk in hand. In the back row sit two ghosts: Monge<\/strong> (with a ruler) and Kantorovich<\/strong> (with a ledger).<\/em><\/p>\n\n\n\n

    \n
    \"\"<\/figure>\n<\/figure>\n\n\n\n

    Brenier (calm, to the room):<\/strong> Suppose our cost is the squared distance,
    \\[
    c(x,y)=\\frac{1}{2}\\|x-y\\|^2,
    \\]
    and the country\u2019s wheat is not piled into atoms but spread with a density (i.e., \\(\\mu\\) is absolutely continuous). Then the optimal transport is not merely a schedule\u2014it is a map<\/strong>. More precisely:<\/p>\n\n\n\n

    Brenier\u2019s Theorem (1991)<\/strong>. Let \\(\\mu,\\nu\\) be probability measures on \\(\\mathbb{R}^d\\) with finite second moments, and assume \\(\\mu\\ll \\mathrm{Leb}\\) (no atoms). For \\(c(x,y)=\\tfrac12|x-y|^2\\), there exists a (a.e.\\ unique) optimal Monge map
    \\[
    \\boxed{\\,T=\\nabla \\Phi\\,}
    \\]
    for some convex potential \\(\\Phi:\\mathbb{R}^d\\to\\mathbb{R}\\cup{+\\infty}\\), and the optimal plan is concentrated on its graph:
    \\[
    \\pi^\\ast=(\\mathrm{id},T)_{\\#}\\mu .
    \\]<\/p>\n\n\n\n

      \n
    1. Finite second moments<\/em>: \\(\\int |x|^2\\,d\\mu(x)<\\infty\\) and \\(\\int |y|^2\\,d\\nu(y)<\\infty\\). This says the average squared distance of mass from the origin is finite (a finite “moment of inertia”). It guarantees that the quadratic transport cost \\(\\int \\tfrac12|x-y|^2\\,d\\pi\\) can be finite for some coupling \\(\\pi\\).<\/li>\n\n\n\n
    2. Absolute continuity of \\(\\mu\\):<\/em> no single point of \\(X\\) carries positive mass. Physically, the supply is a density, not a pile of atoms. This prevents a point-mass from being forced to split and allows the optimal coupling to be given by a function <\/em>\\(T\\) rather than a multivalued plan.<\/li>\n\n\n\n
    3. Quadratic cost<\/em>: the geometry of \\(\\tfrac12|x-y|^2\\) forces optimal couplings to be (c)-cyclically monotone; such sets are subdifferentials of convex functions. This is the mechanism that produces \\(T=\\nabla\\Phi\\).<\/li>\n<\/ol>\n\n\n\n

      Brenier (sketches gradient arrows<\/strong>): Think of \\(\\Phi\\) as a convex landscape over the country. At location \\(x\\), the arrow \\(\\nabla\\Phi(x)\\) points “straight uphill.” My theorem says: for squared-distance cost and a smooth supply, the cheapest rearrangement sends each infinitesimal grain along these gradient arrows to its unique destination \\(T(x)\\). Convexity forbids cost-lowering cycles; gradients do not swirl.
      Each farm-point \\(x\\) sends its infinitesimal mass along \\(\\nabla\\Phi(x)\\) to a unique mill \\(T(x)\\). No crossings, no marathons.<\/p>\n\n\n\n

      Monge (ghost, delighted)<\/strong>: Une fl`eche par point<\/em>. The arrows return!

      \ud83e\uddee Kantorovich (ghost, approving)<\/strong>: Through a convex potential\u2014my ledger smiles.<\/p>\n\n\n\n

      Consequences (when densities exist)<\/strong>.
      If \\(\\mu=\\rho(x)\\,dx\\) and \\(\\nu=\\sigma(y)\\,dy\\), then \\(\\Phi\\) solves, in the weak (a.e.) sense, the Monge–Amp`ere equation
      \\[
      \\det\\big(\\nabla^2\\Phi(x)\\big)=\\frac{\\rho(x)}{\\sigma\\big(\\nabla\\Phi(x)\\big)}\\,,
      \\qquad\\text{and}\\qquad
      T_{\\#}\\mu=\\nu .
      \\]
      The displacement interpolation
      \\[
      \\mu_t:=\\big((1-t)\\,\\mathrm{id}+t\\,T\\big)_{\\#}\\mu,\\qquad t\\in[0,1],
      \\]
      is a constant-speed geodesic in the 2-Wasserstein space \\(W_2(\\mathbb{R}^d)\\) between \\(\\mu\\) and \\(\\nu\\).<\/p>\n\n\n\n

      Short answer to why.<\/strong>
      Optimal couplings for the quadratic cost are (c)-cyclically monotone; Rockafellar\u2019s theorem puts such sets inside the subdifferential \\(\\partial\\Phi\\) of a convex \\(\\Phi\\). Absolute continuity of \\(\\mu\\) makes \\(\\partial\\Phi\\) single-valued \\(\\mu\\)-a.e., yielding \\(T=\\nabla\\Phi\\) (existence and uniqueness).<\/p>\n\n\n\n

      Brenier drops the chalk. The ghosts exchange a courteous bow: arrows for Louis XVI, convexity for the Bureau. Outside, somewhere, peasants are happy, and there is enough bread to feed the kingdom.<\/em><\/p>\n\n\n\n

      Acknowledgments<\/strong>. I\u2019m grateful to professor Nestor Gullen<\/a>, who introduced me to the main ideas of optimal transport during my time as a volunteer assistant on his Algorithms for Convex Surface Construction <\/em><\/strong>project. Many thanks to research fellows Stephanie Atherton; Matthew Hellinger; Minghao Ji; Amar KC; Marina Oliveira Levay Reis, for being a joy to work with\u2014curious, rigorous, and creative, and to professor Justin Solomon<\/a> as well for assigning me this project, and making SGI awesome.<\/p>\n\n\n\n

      Historical note & disclaimer<\/h5>\n\n\n\n

      This tale is fictional<\/strong>. The conversations attributed to Louis XVI, Gaspard Monge, and Leonid Kantorovich are the product of the author\u2019s imagination.<\/p>\n\n\n\n

      Bibliography<\/strong><\/p>\n\n\n\n

        \n
      1. Introduction to Gradient Flows in the 2-Wasserstein Space | Abdul Fatir Ansari<\/em>. (n.d.). Retrieved August 22, 2025, from https:\/\/abdulfatir.com\/blog\/2020\/Gradient-Flows\/<\/a><\/li>\n\n\n\n
      2. Frederic Schuller (Director). (2016, March 13). Measure Theory  -Lec05- Frederic Schuller<\/em> [Video recording]. https:\/\/www.youtube.com\/watch?v=6ad9V8gvyBQ<\/a><\/li>\n\n\n\n
      3. Monge, G. (1781). M\u00e9moire sur la th\u00e9orie des d\u00e9blais et des remblais. Mem. Math. Phys. Acad. Royale Sci.<\/em>, 666\u2013704.<\/li>\n\n\n\n
      4. Peyr\u00e9, G. (2025). Optimal Transport for Machine Learners. arXiv Preprint arXiv:2505.06589<\/em>.<\/li>\n\n\n\n
      5. Santambrogio, F. (n.d.). Optimal Transport for Applied Mathematicians \u2013 Calculus of Variations, PDEs and Modeling<\/em>.

        <\/li>\n<\/ol>\n\n\n\n


        <\/p>\n\n\n\n


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        \ud83c\udfad A Tale of Two Transports: A Comic Drama in Three Acts \ud83c\udfad Dramatis Personae Act I \u2014 A Village in France, 1780 Scene: Grey dawn. Carts piled with wheat creak by; a boy staggers under chiming milk pails; two men shoulder grapes toward a wine press heroically far from grapes. Chickens run quality assurance […]<\/p>\n","protected":false},"author":80,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[52,59],"ppma_author":[57],"class_list":["post-1419","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-math","tag-optimal-transport"],"authors":[{"term_id":57,"user_id":80,"is_guest":0,"slug":"ehsan-ali","display_name":"Ehsan Shams","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/4926607916c31d069143cd9b87b4ca580fcc97287349ad6c1767be4f96a9d062?s=96&d=mm&r=g","0":null,"1":"","2":"","3":"","4":"","5":"","6":"","7":"","8":""}],"_links":{"self":[{"href":"https:\/\/summergeometry.org\/sgi2025\/wp-json\/wp\/v2\/posts\/1419","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/summergeometry.org\/sgi2025\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/summergeometry.org\/sgi2025\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2025\/wp-json\/wp\/v2\/users\/80"}],"replies":[{"embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2025\/wp-json\/wp\/v2\/comments?post=1419"}],"version-history":[{"count":10,"href":"https:\/\/summergeometry.org\/sgi2025\/wp-json\/wp\/v2\/posts\/1419\/revisions"}],"predecessor-version":[{"id":2031,"href":"https:\/\/summergeometry.org\/sgi2025\/wp-json\/wp\/v2\/posts\/1419\/revisions\/2031"}],"wp:attachment":[{"href":"https:\/\/summergeometry.org\/sgi2025\/wp-json\/wp\/v2\/media?parent=1419"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2025\/wp-json\/wp\/v2\/categories?post=1419"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2025\/wp-json\/wp\/v2\/tags?post=1419"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/summergeometry.org\/sgi2025\/wp-json\/wp\/v2\/ppma_author?post=1419"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}