The sharp Whitney extension theorem for convex $C^1$ Lipschitz functions
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866917251138977792 |
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| author | Mudarra, Carlos |
| author_facet | Mudarra, Carlos |
| contents | For an arbitrary set $E \subset \mathbb{R}^n$, and functions $f:E \to \mathbb{R}$, $G: E\to \mathbb{R}^n$ with $G$ bounded, we construct $C^1(\mathbb{R}^n)$ convex extensions $(F, \nabla F)$ of $(f,G)$ with the sharp Lipschitz constant
$$
\mathrm{Lip}(F) = \sup_{x\in E} |G(x)|,
$$
provided that $(f,G)$ satisfies the pertinent necessary and sufficient conditions for $C^1$ convex, and Lipschitz extendability. Also, these extensions can be constructed with prescribed global behavior in terms of directions of coercivity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05642 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The sharp Whitney extension theorem for convex $C^1$ Lipschitz functions Mudarra, Carlos Classical Analysis and ODEs Functional Analysis For an arbitrary set $E \subset \mathbb{R}^n$, and functions $f:E \to \mathbb{R}$, $G: E\to \mathbb{R}^n$ with $G$ bounded, we construct $C^1(\mathbb{R}^n)$ convex extensions $(F, \nabla F)$ of $(f,G)$ with the sharp Lipschitz constant $$ \mathrm{Lip}(F) = \sup_{x\in E} |G(x)|, $$ provided that $(f,G)$ satisfies the pertinent necessary and sufficient conditions for $C^1$ convex, and Lipschitz extendability. Also, these extensions can be constructed with prescribed global behavior in terms of directions of coercivity. |
| title | The sharp Whitney extension theorem for convex $C^1$ Lipschitz functions |
| topic | Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2602.05642 |