Convex $C^1$ extensions of $1$-jets from compact subsets of Hilbert spaces

Daniel Azagra; Carlos Mudarra

doi:10.5802/crmath.62

Functional Analysis

Convex

C^{1}

extensions of

1

-jets from compact subsets of Hilbert spaces

Daniel Azagra ¹; Carlos Mudarra ²

¹ ICMAT (CSIC-UAM-UC3-UCM), Departamento de Análisis Matemático y Matemática Aplicada, Facultad Ciencias Matemáticas, Universidad Complutense, 28040, Madrid, Spain
² Aalto University, Department of Mathematics and Systems Analysis, P.O. BOX 11100, FI-00076 Aalto, Finland

Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 551-556

Abstract (Original language)
Résumé

Let $X$ denote a Hilbert space. Given a compact subset $K$ of $X$ and two continuous functions $f : K \to ℝ$ , $G : K \to X$ , we show that a necessary and sufficient condition for the existence of a convex function $F \in C^{1} (X)$ such that $F = f$ on $K$ and $\nabla F = G$ on $K$ is that the $1$ -jet $(f, G)$ satisfies:

(1) $f (x) \geq f (y) + 〈 G (y), x - y 〉$ for all $x, y \in K$ , and
(2) if $x, y \in K$ and $f (x) = f (y) + 〈 G (y), x - y 〉$ then $G (x) = G (y)$ .

We also solve a similar problem for $K$ replaced with an arbitrary bounded subset of $X$ , and for $C^{1} (X)$ replaced with the class $C_{b}^{1, u} (X)$ of differentiable functions with uniformly continuous derivatives on bounded subsets of $X$ .

Soit $X$ un espace de Hilbert. Nouns montrons que, étant donné un sous-ensemble compact $K$ de $X$ et deux fonctions continues $f : K \to ℝ$ , $G : K \to X$ , pour qu’il existe une fonction convexe $F \in C^{1} (X)$ telle que $(F, \nabla F) = (f, G)$ dans $K,$ il faut et il suffit que

(1) $f (x) \geq f (y) + 〈 G (y), x - y 〉$ pour tout $x, y \in K$ , et que
(2) si $x, y \in K$ et $f (x) = f (y) + 〈 G (y), x - y 〉$ , $G (x) = G (y)$ .

Nous résolvons également un problème similaire pour $K$ remplacé par un sous-ensemble borné arbitraire de $X$ , et pour $C^{1} (X)$ remplacé par la classe $C_{b}^{1, u} (X)$ de fonctions différentiables avec des dérivées uniformément continues sur les sous-ensembles bornés de $X$ .

Received: 2019-12-03
Revised: 2020-04-02
Accepted: 2020-05-05
Published online: 2020-09-14

DOI: 10.5802/crmath.62

Classification: 26B05, 26B25, 52A05, 52A20

Author's affiliations:

Daniel Azagra ¹; Carlos Mudarra ²

¹ ICMAT (CSIC-UAM-UC3-UCM), Departamento de Análisis Matemático y Matemática Aplicada, Facultad Ciencias Matemáticas, Universidad Complutense, 28040, Madrid, Spain
² Aalto University, Department of Mathematics and Systems Analysis, P.O. BOX 11100, FI-00076 Aalto, Finland

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2020__358_5_551_0,
     author = {Daniel Azagra and Carlos Mudarra},
     title = {Convex $C^1$ extensions of $1$-jets from compact subsets of {Hilbert} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {551--556},
     year = {2020},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {5},
     doi = {10.5802/crmath.62},
     language = {en},
}

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Daniel Azagra; Carlos Mudarra. Convex $C^1$ extensions of $1$-jets from compact subsets of Hilbert spaces. Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 551-556. doi: 10.5802/crmath.62

References
Cited by

[1] Daniel Azagra; Erwan Le Gruyer; Carlos Mudarra Explicit formulas for $C^{1, 1}$ and $C_{conv}^{1, ω}$ extensions of 1-jets in Hilbert and superreflexive spaces, J. Funct. Anal., Volume 274 (2018) no. 10, pp. 3003-3032 | Zbl | MR | DOI

[2] Daniel Azagra; Carlos Mudarra Whitney Extension Theorems for convex functions of the classes $C^{1}$ and $C^{1, ω}$ , Proc. Lond. Math. Soc., Volume 114 (2017) no. 1, pp. 133-158 | Zbl | MR | DOI

[3] Daniel Azagra; Carlos Mudarra Global geometry and $C^{1}$ convex extensions of 1-jets, Anal. PDE, Volume 12 (2019) no. 4, pp. 1065-1099 | Zbl | DOI | MR

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