{"ID":2921674,"CreatedAt":"2026-06-02T02:42:49.606572591Z","UpdatedAt":"2026-06-03T05:56:00.181519634Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.01154","arxiv_id":"2606.01154","title":"A convexity criterion via the De Giorgi slope","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 $f$.","short_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\\ma...","url_abs":"https://arxiv.org/abs/2606.01154","url_pdf":"https://arxiv.org/pdf/2606.01154v1","authors":"[\"Tahar Z. Boulmezaoud\",\"Aris Daniilidis\",\"Trí Minh Lê\"]","published":"2026-05-31T10:48:41Z","proceeding":"math.FA","tasks":"[\"math.FA\",\"math.OC\"]","methods":"[]","has_code":false}