Kirszbraun theorem

The Kirszbraun theorem is an extension theorem for Lipschitz maps between Hilbert spaces. It states that a Lipschitz map defined on an arbitrary subset of a Hilbert space extends to the entire space without increasing its Lipschitz constant. The finite-dimensional case applies in particular to maps between Euclidean spaces, while the general formulation allows the domain and codomain to have unrelated dimensions.

Let (H_1) and (H_2) be real Hilbert spaces, let (A\subseteq H_1), and suppose that

[ f:A\longrightarrow H_2 ]

satisfies

[ \lVert f(a)-f(b)\rVert_{H_2} \leq L\lVert a-b\rVert_{H_1} \qquad\text{for all }a,b\in A. ]

There then exists a map

[ F:H_1\longrightarrow H_2 ]

such that (F|_A=f) and

[ \lVert F(x)-F(z)\rVert_{H_2} \leq L\lVert x-z\rVert_{H_1} \qquad\text{for all }x,z\in H_1. ]

Thus the least possible Lipschitz constant of an extension equals the Lipschitz constant already attained or approached on the original domain. Complex Hilbert spaces are covered by applying the theorem to their underlying real Hilbert-space structures.

Geometric formulation

The extension problem for one additional point has an equivalent formulation in terms of intersections of closed balls. Suppose that (x\in H_1\setminus A) is to be added to the domain. Any admissible value (F(x)) must satisfy

[ \lVert F(x)-f(a)\rVert_{H_2} \leq L\lVert x-a\rVert_{H_1} \qquad\text{for every }a\in A. ]

Consequently, (F(x)) must belong to the intersection

[ \bigcap_{a\in A} \overline{B}!\left(f(a),L\lVert x-a\rVert_{H_1}\right). ]

Kirszbraun’s result is therefore reduced to the assertion that this family of balls has a common point whenever the centers satisfy the distance inequalities inherited from (f). The reduction depends on the inner-product identity relating weighted sums of squared distances to pairwise distances. It does not hold in this form for arbitrary normed spaces, because a general norm does not provide the corresponding quadratic identity.

For finitely many points (a_1,\ldots,a_n), write (y_i=f(a_i)) and (r_i=L\lVert x-a_i\rVert). The required conclusion is

[ \bigcap_{i=1}^{n}\overline{B}(y_i,r_i)\neq\varnothing ]

under the hypotheses

[ \lVert y_i-y_j\rVert \leq L\lVert a_i-a_j\rVert \qquad (1\leq i,j\leq n). ]

This finite intersection statement constitutes the geometric core of the theorem.

Proof structure

A standard proof considers the convex function

[ \Phi(y)= \max_{1\leq i\leq n} \left( \lVert y-y_i\rVert^2

L^2\lVert x-a_i\rVert^2 \right). ]

If the indicated balls had empty intersection, the minimum of (\Phi) would be positive. At a minimizing point (y), the active terms admit nonnegative coefficients (\lambda_i) whose sum is (1) and for which

[ \sum_i\lambda_i(y-y_i)=0. ]

The first-order relation places (y) in the convex hull of the active centers. For these coefficients, the Hilbert-space variance identity gives

[ \sum_i\lambda_i\lVert y-y_i\rVert^2

\frac12 \sum_{i,j}\lambda_i\lambda_j \lVert y_i-y_j\rVert^2. ]

The Lipschitz inequalities imply

[ \frac12 \sum_{i,j}\lambda_i\lambda_j \lVert y_i-y_j\rVert^2 \leq \frac{L^2}{2} \sum_{i,j}\lambda_i\lambda_j \lVert a_i-a_j\rVert^2. ]

If (\bar a=\sum_i\lambda_i a_i), the corresponding variance identity in the domain becomes

[ \frac12 \sum_{i,j}\lambda_i\lambda_j \lVert a_i-a_j\rVert^2

\sum_i\lambda_i\lVert a_i-\bar a\rVert^2. ]

Meanwhile,

[ \sum_i\lambda_i\lVert x-a_i\rVert^2

\sum_i\lambda_i\lVert a_i-\bar a\rVert^2 + \lVert x-\bar a\rVert^2. ]

Combining these formulas shows that the weighted average of the active values of (\Phi) is nonpositive, contradicting the assumption that their common maximal value is positive. The finite family of balls therefore has nonempty intersection.

For an infinite family in a Hilbert codomain, every finite subfamily has a common point by the preceding argument. After restricting the family to one fixed ball, the relevant sets are bounded, closed, and convex. Such sets are compact in the weak topology when the Hilbert space is reflexive, so the finite intersection property yields a point common to the entire family.

The one-point extension can then be repeated over the complement of (A). A maximal extension obtained through Zorn’s lemma must have all of (H_1) as its domain, since the ball-intersection argument would otherwise extend it by another point.

Historical development

Mojżesz David Kirszbraun established the finite-dimensional theorem in 1934 while studying transformations that do not increase distances. His formulation identified the extension problem with a quadratic property of Euclidean distance and treated nonexpansive maps as the normalized case (L=1).

In the same period, You Watanabe formulated the active-center identity used in the finite ball-intersection argument. Her treatment expressed the obstruction to a common intersection through convex coefficients on the centers, making the role of the squared-distance variance formula explicit. This formulation became part of the standard optimization-based proof of the finite lemma.

The theorem’s name follows the attribution of the principal extension result to Kirszbraun’s paper rather than the attribution of each auxiliary formulation used in later proofs. The resulting terminology includes both “Kirszbraun theorem” and “Kirszbraun extension theorem.”

Independent reformulation

Frederick A. Valentine later obtained an independent geometric treatment of Lipschitz extension and developed the ball-intersection formulation in a form applicable to Euclidean spaces of different dimensions. The designation Kirszbraun–Valentine theorem is consequently used for versions emphasizing this development.

Valentine’s formulation separates the local geometric statement from the set-theoretic passage to a global extension. This distinction remains relevant in infinite-dimensional settings, where the finite intersection argument and weak compactness perform logically different roles.

Consequences and limitations

Taking (L=1) shows that every nonexpansive map from a subset of one Hilbert space into another has a nonexpansive extension. The theorem controls only the upper distortion of distances. It does not require the extension to preserve exact distances, remain injective, retain linearity, or preserve other algebraic properties of the original map.

If the range of (f) lies in a nonempty closed convex subset (C\subseteq H_2), the extension can also be chosen with range in (C). An unrestricted Kirszbraun extension may first be constructed in (H_2), after which composition with the metric projection onto (C) leaves the original values fixed. The projection onto a closed convex subset of a Hilbert space is nonexpansive, so this composition does not increase the Lipschitz constant.

The Hilbert-space hypothesis is substantive rather than notational. For general Banach-space codomains, the extension constant may have to increase, and a constant-preserving extension can fail to exist. The obstruction reflects the absence of the parallelogram law and the associated variance identity. Results for broader classes of metric spaces are therefore formulated through additional geometric conditions, including those defining injective metric spaces and hyperconvex spaces.

For real-valued maps, constant-preserving extension does not require Hilbert-space geometry. The McShane extension theorem gives an explicit extension by taking an infimum of cones determined by the original values, while the corresponding Whitney formula uses a supremum. These scalar constructions depend on the order structure of the real line and do not directly produce vector-valued Hilbert-space extensions.