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 n (not necessarily bounded) and f : E , G : E n be functions. We provide necessary and sufficient conditions for the 1-jet (f,G) to have an extension (F,F) with F : n convex and C1 . 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 C1 with prescribed normals at the points of an arbitrary subset of n .

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