#### Vol. 12, No. 4, 2019

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Global geometry and $C^1$ convex extensions of 1-jets

### Daniel Azagra and Carlos Mudarra

Vol. 12 (2019), No. 4, 1065–1099
DOI: 10.2140/apde.2019.12.1065
##### Abstract

Let $E$ be an arbitrary subset of ${ℝ}^{n}$ (not necessarily bounded) and $f:E\to ℝ$, $G:E\to {ℝ}^{n}$ be functions. We provide necessary and sufficient conditions for the $1$-jet $\left(f,G\right)$ to have an extension $\left(F,\nabla F\right)$ with $F:{ℝ}^{n}\to ℝ$ convex and ${C}^{1}$. Additionally, if $G$ is bounded we can take $F$ so that $Lip\left(F\right)\lesssim \parallel G{\parallel }_{\infty }$. As an application we also solve a similar problem about finding convex hypersurfaces of class ${C}^{1}$ with prescribed normals at the points of an arbitrary subset of ${ℝ}^{n}$.

##### Keywords
convex function, $C^1$ function, Whitney extension theorem, global differential geometry, differentiable function
##### Mathematical Subject Classification 2010
Primary: 26B05, 26B25, 52A20