{
  "abstract": "Let $X$ be a Banach space and $f\\in\\mathcal{C}^1(X)$ be bounded from below. We show that if for some $m\\geq 1$, the function $x\\mapsto \\|\\nabla f(x)\\|^m$ is convex, then $f$ is convex. We also establish a more general version of this result: if $f$ is continuous and bounded from below, then it is convex, provided $x\\mapsto s_f(x)^m$ is convex for some $m\\geq 1$, where $s_f$ denotes the (De Giorgi) metric slope of $f$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.01154",
  "author_total": 3,
  "authors": [
    {
      "name": "Tahar Z. Boulmezaoud"
    },
    {
      "name": "Aris Daniilidis"
    },
    {
      "name": "Trí Minh Lê"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/carwfzatfrna5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.01154",
  "fulltext_url": null,
  "id": "carwfzatfrna5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.01154",
  "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",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7163165954",
  "pdf_url": "https://arxiv.org/pdf/2606.01154",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/carwfzatfrna5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2606.01154",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.01154"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A convexity criterion via the De Giorgi slope",
  "title_text": "A convexity criterion via the De Giorgi slope",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/carwfzatfrna5",
  "venue": "arXiv",
  "versions": []
}