Minimal Lipschitz Extensions to Differentiable Functions Defined on a Hilbert Space

Abstract

We generalize the Lipschitz constant to fields of affine jets and prove that such a field extends to a field of total domain R n with the same constant. This result may be seen as the analog for fields of the minimal Kirszbraun's extension theorem for Lipschitz functions and, therefore, establishes a link between Kirszbraun's theorem and Whitney's theorem. In fact this result holds not only in Euclidean R n but also in general (separable or not) Hilbert space. We apply the result to the functional minimal Lipschitz differentiable extension problem in Euclidean spaces and we show that no Brudnyi-Shvartsman-type theorem holds for this last problem. We conclude with a first approach of the absolutely minimal Lipschitz extension problem in the differentiable case which was originally studied by Aronsson in the continuous case.

Kirszbraun’s Theorem via an Explicit Formula

“…(3) If c > 0, then G is strongly biLipschitz with SBilip(G) ≥ 2 c+M min{1, cM }. (4) For c = −M we recover Wells's condition W 1,1 considered in [24,18,3]. For c = 0, (SCW 1,1 ) is just condition (CW 1,1 ) of [2,3].…”

“…See also [1], where some of these techniques are used to construct definable versions of Helly's and Kirszbraun theorems in arbitrary definably complete expansions of ordered fields. Finally, in 2015 E. Le Gruyer and T-V. Phan provided sup-inf explicit extension formulas for Lipschitz mappings between finite dimensional spaces by relying on Le Gruyer's solution to the minimal C 1,1 extension problem for 1-jets; see [19,Theorem 32 and 33] and [18]. In this note we present a short proof of the Kirszbraun-Valentine theorem in which the extension is given by an explicit formula.…”

“…for all x, y ∈ E. In [24,18] it was proved that condition (W 1,1 ) with constant M > 0 is a necessary and sufficient condition on f : E → R, G : E → X for the existence of a function F ∈ C 1,1 (X) with Lip(∇F ) ≤ M and such that F = f and ∇F = G on E. More recently, as a consequence of a similar extension theorem for C 1,1 convex functions, we have found an explicit formula for such an extension F.…”

Global geometry and C1 convex extensions of 1-jets

“…Besides the very basic character of Problem 1.1, there are other reasons for wanting to solve this kind of problem, as extension techniques for convex functions have natural applications in Analysis, Differential Geometry, PDE theory (in particular Monge-Ampère equations), Economics, and Quantum Computing. See the introductions of [5,15,28] for background about convex extensions problems, and see [7,10,11,12,13,14,18,21,23] and the references therein for information about general Whitney extension problems. Let C 1 conv (R n ) stand for the set of all functions f : R n → R which are convex and of class C 1 .…”

Explicit formulas for C1,1 and Cconv1,ω extensions of 1-jets in Hilbert and superreflexive spaces

“…Recall also that the function F constructed in this way has the property that Lip(∇F ) ≤ k(n)M, where k(n) is a constant depending only on n (but going to infinity as n → ∞), and Lip(∇F ) denotes the Lipschitz constant of the gradient ∇F . In [28,20] it was shown, by very different means, that this C 1,1 version of the Whitney extension theorem holds true if we replace R n with any Hilbert space and, moreover, there is an extension operator (f, G) → (F, ∇F ) which is minimal, in the following sense. Given a Hilbert space X with norm denoted by · , a subset E of X, and functions f : E → R, G : E → X, a necessary and sufficient condition for the 1-jet (f, G) to have a C 1,1 extension (F, ∇F ) to the whole space X is that is the trace seminorm of the jet (f, G) on E; see [20] and [21,Lemma 15].…”