Analysis & PDE

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

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

Daniel Azagra and Carlos Mudarra

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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 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 n .

Article information

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

Dates
Received: 4 September 2017
Revised: 13 March 2018
Accepted: 30 July 2018
First available in Project Euclid: 30 October 2018

Permanent link to this document
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

Subjects
Primary: 26B05: Continuity and differentiation questions 26B25: Convexity, generalizations 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45]

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

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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