{
  "abstract": "We investigate the relationship between generalized (q ′ , p ′ )-Poincaré inequalities and transport-divergence inequalities involving χ q -type divergences. Extending the classical correspondence between the (2, 2)-Poincaré inequality and quadratic transport-χ 2 inequalities, we prove that a (q ′ , p ′ )-Poincaré inequality is equivalent to a transportdivergence inequality for conjugates (p, p ′ ) and (q, q ′ ). We then study the convex counterpart of these inequalities. Using weak barycentric optimal transport and convex order, we establish an equivalence between convex (q ′ , p ′ )-Poincaré inequalities and weak transport-divergence inequalities involving a projected χ q divergence. We further characterize the minimizers arising in this projection problem through a geometric condition on the relative density. In dimension one, we analyze the geometric consequences of (q ′ , p ′ )-Poincaré inequalities by identifying sharp conditions on the behavior of the density both at infinity and near isolated zeros. We show that when q ′ > p ′ , the concentration behavior becomes polynomial rather than exponential at infinity. Finally, we interpret these inequalities through the lens of optimization in Wasserstein space. In particular, the equivalence between transport-divergence and Poincaré inequalities appears as an analogue of the equivalence between quadratic growth and Polyak-Lojasiewicz inequalities in finite-dimensional optimization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "name": "Thibaut Le Gouic"
    },
    {
      "name": "Magali Tournus"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/07zwvt5r09q0m.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "07zwvt5r09q0m",
  "issns": [],
  "landing_url": "https://hal.science/hal-05780607v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05780607/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/07zwvt5r09q0m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:49Z",
      "native_id": "hal-05780607",
      "source": "hal",
      "url": "https://hal.science/hal-05780607v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Generalized and Convex Poincaré Inequalities via Transport-Divergence Inequalities",
  "title_text": "Generalized and Convex Poincaré Inequalities via Transport-Divergence Inequalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:49Z",
  "url": "https://research.draconitic.com/paper/07zwvt5r09q0m",
  "venue": "HAL",
  "versions": []
}