Kirszbraun's theorem via an explicit formula

Let $X,Y$ be two Hilbert spaces, $E$ a subset of $X$ and $G: E \to Y$ a Lipschitz mapping. A famous theorem of Kirszbraun's states that there exists $\widetilde{G} : X \to Y$ with $\widetilde{G}=G$ on $E$ and $\textrm{Lip}(\widetilde{G})=\textrm{Lip}(G).$ In this note we show that in fact the function $$\widetilde{G}:=\nabla_Y(\textrm{conv}(g))( \cdot , 0), \qquad \text{where} $$ $$ g(x,y) = \inf_{z \in E} \lbrace \langle G(z), y \rangle + \tfrac{M}{2} \|(x-z,y)\|^2 \rbrace + \tfrac{M}{2}\|(x,y)\|^2, $$ defines such an extension. We apply this formula to get an extension result for {\em strongly biLipschitz homeomorphisms.} Related to the latter, we also consider extensions of $C^{1,1}$ strongly convex functions.


An explicit formula for Kirszbraun's theorem
In 1934 M.D. Kirszbraun [23] proved that, for every subset E of R n and every Lipschitz function f : E → R m , there exists a Lipschitz extension F : R n → R m of f such that Lip(F ) = Lip(f ). Here Lip(ϕ) denotes the Lipschitz constant of ϕ, that is, x − y .
This theorem was generalized for Hilbert spaces X, Y in place of R n and R m by F.A. Valentine [30] in 1945, and the result is often referred to as the Kirszbraun-Valentine theorem. The proof is rather nonconstructive, in the sense that it requires to use Zorn's lemma or transfinite induction at least in the nonseparable case. In the separable case the proof can be made by induction, considering a dense sequence {x k } in X and at each step managing to extend f from E ∪ {x 1 , . . . , x m } to E ∪ {x 1 , . . . , x m+1 } while preserving the Lipschitz constant of the extension by using Helly's theorem or intersection properties of families of balls, but still it is not clear what the extension looks like. Several other proofs and generalizations which are not constructive either have appeared in the literature; see [26,21,15,11,27,5,1]; apart from Zorn's lemma or induction these proofs are based on intersection properties of arbitrary families of balls, or on maximal extensions of non-expansive operators and Fitzpatrick functions. In 2008 H.H. Bauschke and X. Wang [9] gave the first constructive proof of the Kirszbraun-Valentine theorem of which we are aware; they relied on their previous work [8] on extension and representation of monotone operators and the Fitzpatrick function. See also [2], where some of these techniques are used to construct definable versions of Helly's and Kirszbraun's 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 [25,Theorem 32 and 33] and [24].
In this note we present a short proof of the Kirszbraun-Valentine theorem in which the extension is given by an explicit formula. This proof is based on our previous work concerning C 1,1 extensions of 1-jets with optimal Lipschitz constants of the gradients [4]. See [14] for an alternative construction of such C 1,1 extensions on the Hilbert space, and [12,16,17,18] for the much more difficult question of extending functions (as opposed to jets) to C 1,1 or C m,1 functions on R n .
If X is a Hilbert space, E ⊂ X is an arbitrary subset and (f, G) : E → R × X is a 1-jet on E, we will say that (f, G) satisfies condition (W 1,1 ) with constant M > 0 on E provided that for all x, y ∈ E. In [32,24] 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. Here ∇F (x) denotes the gradient of F at the point x, that is, the unique vector ∇F (x) ∈ X for which DF (x)(v) = ∇F (x), v for every v ∈ X, where DF (x) ∈ X * denotes the Fréchet derivative of F at the point x. 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. Theorem 3.4] Let E be a subset of a Hilbert space X. Given a 1-jet (f, G) satisfying condition (W 1,1 ) with constant M on E, the formula Here conv(g) denotes the convex envelope of g, defined by (2) conv(g)(x) = sup{h(x) : h is convex, proper and lower semicontinuous, h ≤ g}.
Another expression for conv(g) is given by and also by the Fenchel biconjugate of g, that is, see [13,Proposition 4.4.3] for instance. In the case that X is finite dimensional, say X = R n , the expression (3) can be made simpler: by using Carathéodory's Theorem one can show that it is enough to consider convex combinations of at most n + 1 points. That is to say, if g : R n → R then see [28,Corollary 17.1.5] for instance.
In general, the convex envelope does not preserve smoothness of orders higher than C 1 and C 1,1 . For instance, the function g(x, y) = x 2 + e −y 2 defined on R 2 is real analytic and its convex envelope is conv(g)(x, y) = |x| for every (x, y) ∈ R 2 ; see [10]. In [22], Kirchheim and Kristensen proved that the convex envelope of a differentiable function g on R n is of class C 1 , provided that g is coercive. On the other hand, if g is of class C 1,1 on a Hilbert space X, or even if g only satisfies for some M > 0, then conv(g) is of class C 1,1 and Lip(∇ conv(g)) ≤ M ; see [4,Theorem 2.3]. In particular, if g = inf i (g i ) is the infimum of an arbitrary family of parabolas g i , whose second derivatives are uniformly bounded by a constant M > 0, then conv(g) is of class C 1,1 with Lip(∇ conv(g)) ≤ M, provided that g has a convex lower semicontinuous minorant. However, conv(g) is not necessarily of class C 2 even when g is the minimum of two parabolas: if we define g( Theorem 2 (Kirszbraun's theorem via an explicit formula). Let X, Y be two Hilbert spaces, E a subset of X and G : E → Y a Lipschitz mapping. There exists G : Here · X and · Y denote the norm on X and Y respectively. Also, the inner products in X and Y are denoted by ·, · X and ·, · Y respectively. For any function F, ∇ Y F will stand for the Y -partial derivatives of F, that is, the canonical projection from X × Y onto Y composed with ∇F .
Proof. We consider on X × Y the norm given by (x, y) = Then the jet (f * , G * ) satisfies condition (W 1,1 ) on E × {0} with constant M (see inequality (1)). Indeed, by the definition of f * and G * we can write, for every (x, 0), (y, 0) ∈ E × {0}, Therefore, Theorem 1 asserts that the function F defined by Finally, the expressions defining G and g can be simplified as Let X be a Hilbert space with inner product and associated norm denoted by ·, · and · respectively. If E ⊂ X is arbitrary and G : E → X is a mapping, we say that G is firmly non-expansive if Important examples of firmly non-expansive mappings are the metric projections onto closed convex sets of Hilbert spaces and the proximal mappings prox f : X → X of proper lower semicontinuous convex functions f : X → (−∞, +∞]; see [6,Chapter 12]. Firmly non-expansive mappings arise naturally in convex feasibility problems too: given a family C 1 , . . . , C N of closed convex sets of a Hilbert space, find a point x ∈ i C i . Also, these mappings are known to be resolvents J A = (A + I) −1 of monotone or maximally monotone operators A : X ⇒ X, and they play a crucial role in the following basic problem that arises in several branches of applied mathematics: given a maximally monotone operator A : X ⇒ X, find a point x ∈ X with 0 ∈ Ax. For more information about firmly non-expansive mappings and their applications; see [5,6,7,9] and the references therein.
It is well-known that a mapping G : E → X is firmly non-expansive if and only 2G − I : E → X is 1-Lipschitz, where I denotes the identity map; see [6, Proposition 4.2] for a proof of this fact. Using this characterization and Theorem 2 we obtain the following corollary.
Corollary 3. Let G : E → X be a firmly non-expansive mapping defined on a subset E of a Hilbert space X. Then G can be extended to a firmly non-expansive mapping G : X → X by means of the formula

Extensions of strongly biLipschitz mappings
In this section we consider strongly biLipschitz mappings, which appear naturally as derivatives of strongly convex C 1,1 functions, and we provide an extension result for this class of mappings.
Definition 4. Let E be a subset of a Hilbert space X. We say that a mapping G : E → X is strongly biLipschitz provided that Strongly biLipschitz mappings may be interesting in regard to some problems in computer vision or image processing where one needs to match points in R n : for instance, given two sets of points in R n with equal cardinality, find a homeomorphism from R n onto itself which does not distort distances too much and takes one set onto the other. Supposing that the data satisfy the strongly biLipschitz condition, our explicit formula for such an extension can be useful. Also, in [7,Corollary 4.5] it is shown that strongly biLipschitz mappings are closely related to contractive mappings: a maximally monotone mapping G is strongly biLipschitz if and only if its reflected resolvent N = 2(G + I) −1 − I is a contractive mapping. See [29,Chapter 12] for more information about resolvent mappings and maximally monotone operators. It should also be noted that it is not generally true that a biLipschitz mapping whose domain and range is a subset of the same Hilbert space X extends to a total one-to-one continuous mapping, as shown by the following example.
Example 5. Let | · | denote the euclidean norm on R n . Let C = {x ∈ R n : |x| = 1} ∪ {p}, p be any point with |p| > 1, and g : C → R n be defined by g(x) = x for |x| = 1 and g(p) = 0. Then both g and g −1 are Lipschitz but no continuous extension of g to R n can be one-to-one. However, this is true for the class of strongly biLipschitz mappings, and moreover, the extension can be performed without increasing what seems natural to call the strong biLipschitz constant SBilip(G), as we will show by using the extension formula for Lipschitz mappings given by Theorem 2. Proposition 6. If G : E → X is strongly biLipschitz then G is biLipschitz.
Proof. For every x, y ∈ E we have where α = SBilip(G) is as in Definition 4. It follows that G is one-to-one. Note that we can write (7) in the equivalent form Setting λ := G(x) − G(y) x − y for x = y, (7) holds if and only if Therefore G is a biLipschitz mapping.
Remark 7. (i) If X = R, then the strongly biLipschitz functions are exactly the strictly increasing biLipschitz functions.
then G is a strongly biLipschitz function with SBilip(G) = α. However the composition of strongly biLipschitz isometries need not be strongly biLipschitz (for instance, if r : R 2 → R 2 is defined by r(z) = e πi/4 z then r is strongly biLipschitz, but r 2 is not (and in fact r 4 = −id is not strongly biLipschitz locally on any disk).
We obtain from (8) that G is strongly biLipschitz on X with SBilip( G) = α. Also, since T is an extension of G − 1 α I, it is obvious that G is an extension of G.

C 1,1 strongly convex functions
In this section we characterize the 1-jets which can be interpolated by strongly convex functions of class C 1,1 in Hilbert spaces. A function F : X → R is strongly convex if F − c · 2 is convex for some c > 0. In Proposition 10 below we will see that the gradient of a C 1,1 strongly convex function is a biLipschitz mapping. These functions arise naturally when studying smooth manifolds of positive curvature as well as in problems involving Monge-Ampére equations. See the papers [19,20,31] for some results and problems involving smooth strongly convex functions.
Throughout this section X denotes a Hilbert space with norm and inner product denoted by ·, · and · respectively.
Definition 9. Let E ⊆ X be arbitrary, (f, G) : E → R × X be a 1-jet and c ∈ R, M > 0 constants such that M > c. We say that (f, G) satisfies condition (SCW 1,1 ) with constants (c, M ) provided that Proposition 10. Assume that (f, G) : E → R × X satisfies condition (SCW 1,1 ) with constants (c, M ). Then, the following properties hold. (i) For every x, y ∈ E we have (iii) If c > 0, then G is strongly biLipschitz with SBilip(G) ≥ 2 c+M min{1, cM }. (iv) For c = −M we recover Wells's condition W 1,1 considered in [32,24,4]. For c = 0, (SCW 1,1 ) is just condition (CW 1,1 ) of [3,4]. For c ∈ (0, M ] we have what can be called a C 1,1 strongly convex 1-jet, which in the extreme case c = M becomes the restriction of a quadratic function to E. Proof. (i) Let x, y ∈ E. By summing the inequalities , which is equivalent to the desired estimation.
We say that a 1-jet (f, G) : E → R × X satisfies condition (CW 1,1 ) with constant M > 0 on E provided that 2 for every x, y ∈ E. In [4,Theorem 2.4] it was shown that (CW 1,1 ) is a necessary and sufficient condition on (f, G) for the existence of a C 1,1 (X) convex extension F of f with ∇F = G on E. Proof. Assume first that (f, G) satisfies (SCW 1,1 ) with constants (c, M ) on E. We have Proposition 12. Let F ∈ C 1,1 (X) be such that Lip(∇F ) ≤ M and g := F − c 2 · 2 is a convex function, where c ∈ R, M > 0 are two constants (if c > 0, we call such a function F a C 1,1 (globally) strongly convex function). Then the following properties hold. Therefore, according to Definition 9, (F, ∇F ) satisfies condition (SCW 1,1 ) on X with constants (c, M ).

Proof.
(i) and (ii) Given x, h ∈ X, the fact that Lip(∇F ) ≤ M and the convexity of g yield This shows that c ≤ M and, again by convexity of g, that Lip(∇g) ≤ M − c. Finally observe that if M = c, then g is affine and therefore F is a quadratic function.
(iii) By virtue of (ii), the convexity of g implies that (g, ∇g) satisfies condition (CW 1,1 ) with constant M − c on X. Thus Lemma 11 gives the desired inequality.
Theorem 13. Let E be an arbitrary subset of a Hilbert space X, f : E → R, G : E → X be two functions, and c ∈ R, M > 0 be two constants. There exists a function F ∈ C 1,1 (X) such that F = f, ∇F = G on E, Lip(∇F ) ≤ M, and F − c 2 · 2 is a convex function if and only if the jet (f, G) satisfies condition (SCW 1,1 ) with constants (c, M ) on E. In fact, F can be defined by means of the formula F = conv(g) + c 2 · 2 , g(x) = inf y∈E f (y) + G(y), x − y + M 2 x − y 2 − c 2 x 2 , x ∈ X.
Moreover, if H is another function of class C 1,1 (X) satisfying the above properties, then H ≤ F on X.