## Analysis & PDE

• Anal. PDE
• Volume 12, Number 4 (2019), 1065-1099.

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

#### Abstract

Let $E$ be an arbitrary subset of $ℝ n$ (not necessarily bounded) and $f : E → ℝ$, $G : E → ℝ n$ be functions. We provide necessary and sufficient conditions for the $1$-jet $( f , G )$ to have an extension $( F , ∇ F )$ with $F : ℝ n → ℝ$ convex and $C 1$. Additionally, if $G$ is bounded we can take $F$ so that $Lip ( F ) ≲ ∥ G ∥ ∞$. 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 $ℝ n$.

#### Article information

Source
Anal. PDE, Volume 12, Number 4 (2019), 1065-1099.

Dates
Revised: 13 March 2018
Accepted: 30 July 2018
First available in Project Euclid: 30 October 2018

https://projecteuclid.org/euclid.apde/1540864861

Digital Object Identifier
doi:10.2140/apde.2019.12.1065

Mathematical Reviews number (MathSciNet)
MR3869386

Zentralblatt MATH identifier
06991227

#### Citation

Azagra, Daniel; Mudarra, Carlos. Global geometry and $C^1$ convex extensions of 1-jets. Anal. PDE 12 (2019), no. 4, 1065--1099. doi:10.2140/apde.2019.12.1065. https://projecteuclid.org/euclid.apde/1540864861

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