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arXiv · 2602.05642

The sharp Whitney extension theorem for convex $C^1$ Lipschitz functions

Carlos Mudarra
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Abstract

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.

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Carlos Mudarra. 2026-02-05. The sharp Whitney extension theorem for convex $C^1$ Lipschitz functions. https://arxiv.org/abs/2602.05642

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