{
  "abstract": "A contiguous cartogram is a map whose area is proportional to a quantity such as population. The defining condition, that the Jacobian determinant of the deformation equals the density, is one equation for two unknown functions, so every cartogram method adds a tie-breaker, usually implicitly. Optimal transport makes the tie-breaker explicit: among all density-equalising maps of the frame onto itself, take the one that moves the population least in the mean-square sense. By Brenier's theorem that map is the gradient of a convex potential, so it has no local rotation anywhere and cannot fold. We compute this map for the world population of 2025 on a 4096 by 4096 Mercator grid (10 km cells) from the GHS-POP raster, using the fixed-point Monge-Ampère iteration of Benamou, Froese and Oberman with spectral Poisson solves, continuation in the population share and an ocean-only buffer, on a laptop GPU. Against a Gastner-Newman diffusion cartogram of the same density we find the same transport cost to within 0.4 per cent and density errors of a few per cent for both, but a median local rotation of 8.8 degrees for diffusion against 0.01 for transport, and a median anisotropy of 6.30 against 3.93. The construction extends to local refinement, by transporting a city's 100 m population onto the area measure the global map assigns to it, and to a semi-discrete counterpart: 8,192 Laguerre cells of 1.00 million people each. Code, data provenance and every figure's inputs are public.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14089",
  "author_total": 1,
  "authors": [
    {
      "name": "Philipp Bogdan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c9p1vyqzs35fb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14089",
  "fulltext_url": null,
  "id": "c9p1vyqzs35fb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14089",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "astro-ph.EP"
    ],
    [
      "arxiv",
      "cs.CG"
    ],
    [
      "arxiv",
      "cs.GR"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7213369001",
  "pdf_url": "https://arxiv.org/pdf/2609.14089",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c9p1vyqzs35fb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:01Z",
      "native_id": "2609.14089",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14089"
    }
  ],
  "subjects": [
    {
      "field": "astronomy",
      "origin": "arxiv:astro-ph.EP",
      "subfield": "planetary"
    }
  ],
  "title": "The optimal-transport cartogram: world population as a Brenier map",
  "title_text": "The optimal-transport cartogram: world population as a Brenier map",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:01Z",
  "url": "https://research.draconitic.com/paper/c9p1vyqzs35fb",
  "venue": "arXiv",
  "versions": []
}