Global geometry and C1 convex extensions of 1-jets

Azagra, Daniel; Mudarra, Carlos

doi:10.2140/apde.2019.12.1065

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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 $(f, G)$ to have an extension $(F, \nabla F)$ with $F : ℝ^{n} \to ℝ$ convex and $C^{1}$ . Additionally, if $G$ is bounded we can take $F$ so that $Lip (F) ≲ ∥ G ∥_{\infty}$ . 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 $ℝ^{n}$ .

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Keywords

convex function, $C^1$ function, Whitney extension theorem, global differential geometry, differentiable function

Mathematical Subject Classification 2010

Primary: 26B05, 26B25, 52A20

Milestones

Received: 4 September 2017

Revised: 13 March 2018

Accepted: 30 July 2018

Published: 20 October 2018

Authors

Daniel Azagra
ICMAT (CSIC-UAM-UC3-UCM) Departamento de Análisis Matemático Facultad Ciencias Matemáticas Universidad Complutense de Madrid Madrid Spain

Carlos Mudarra
ICMAT (CSIC-UAM-UC3-UCM) Madrid Spain

Vol. 12, No. 4, 2019