Explicit formulas for $C^{1,1}$ Glaeser-Whitney extensions of 1-fields in Hilbert spaces

TL;DR

This paper presents explicit formulas and a simplified proof for the $C^{1,1}$ Glaeser-Whitney extension problem in Hilbert spaces, improving understanding of minimal extensions with practical constructive methods.

Contribution

It offers a new, straightforward proof and explicit formulas for $C^{1,1}$ extensions, nearly achieving minimality in Hilbert spaces, advancing the theory and application of extension problems.

Findings

Provided explicit formulas for $C^{1,1}$ extensions.

Simplified the proof of the extension problem.

Achieved near-minimal extensions up to a factor of (1+√3)/2.

Abstract

We give a simple alternative proof for the $C^{1,1}$--convex extension problem which has been introduced and studied by D. Azagra and C. Mudarra [2]. As an application, we obtain an easy constructive proof for the Glaeser-Whitney problem of $C^{1,1}$ extensions on a Hilbert space. In both cases we provide explicit formulae for the extensions. For the Gleaser-Whitney problem the obtained extension is almost minimal, that is, minimal up to a factor $\frac{1+\sqrt{3}}{2}$ in the sense of Le Gruyer [15].