A convexity criterion via the De Giorgi slope
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913177043730432 |
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| author | Boulmezaoud, Tahar Z. Daniilidis, Aris Lê, Trí Minh |
| author_facet | Boulmezaoud, Tahar Z. Daniilidis, Aris Lê, Trí Minh |
| contents | 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 $f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01154 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A convexity criterion via the De Giorgi slope Boulmezaoud, Tahar Z. Daniilidis, Aris Lê, Trí Minh Functional Analysis Optimization and Control 26B25, 49J52 (Primary) 35F21, 37C10 (Secondary) 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 $f$. |
| title | A convexity criterion via the De Giorgi slope |
| topic | Functional Analysis Optimization and Control 26B25, 49J52 (Primary) 35F21, 37C10 (Secondary) |
| url | https://arxiv.org/abs/2606.01154 |