Lipschitz extension theorems with explicit constants

1.1 Motivation and background

In the following, we denote by X and Y metric spaces, where we call X the domain space and Y the target space. The following general question about the extendability of Lipschitz maps is a natural object of study in metric geometry:

What are conditions on the domain and target space such that any 1-Lipschitz map f : A → Y , where A ⊂ X is an arbitrary subset, can be extended to an L -Lipschitz map F : X → Y ?

Classically, researchers were mainly looking for extensions with L = 1 . Such extensions exist, for example, if the target space is R and the domain space is an arbitrary metric space. Indeed, a short computation reveals that F : X → R defined by

is a 1-Lipschitz extension of f : A → R . This is generally considered the first Lipschitz extension result and was proved by McShane [4], and also independently by Whitney [3, p. 63]. Another classical result with L = 1 is Kirszbraun’s theorem [5,6], where both the target and domain spaces are Hilbert spaces. Also, in this case, the extension can be given by means of an explicit formula (see [7] and references therein).

A generalization of Kirszbraun’s theorem in the setting of Alexandrov geometry was obtained by Lang and Schroeder [8,9]. They impose a lower curvature bound on the domain space and a matching upper curvature bound on the target space. In a similar spirit, by introducing the notions of Markov type and Markov cotype of Banach spaces, another deep generalization of Kirszbraun’s theorem with L > 1 was proved by Ball [10].

Other important results where L > 1 are often obtained in the following setting. Both the domain and target space are arbitrary Banach spaces, but the subset A ⊂ X is assumed to consist of at most n points. It is then proved that L ≤ α ( n ) for some function α : N → R . The first Lipschitz extension result of this flavor was obtained by Marcus and Pisier [11]. They showed that for Hilbert space targets, one can take

if X = L p for some 1 < p < 2 . Johnson and Lindenstrauss [12] proved that for arbitrary domain spaces and Hilbert space targets, one can take α ( n ) = 2 ⋅ log n . We remark that to prove their extension result, Johnson and Lindenstrauss relied on a certain geometric lemma, which is now known under the name Johnson-Lindenstrauss lemma and has found many applications in various areas of mathematics and computer science.

The first result for arbitrary Banach target spaces was obtained by Johnson et al. [13]. They showed that one can take α ( n ) = C ⋅ log n . This was improved by Lee and Naor [1] to

A lower bound on α ( n ) was obtained by Naor and Rabani [14]. Another (very general) lower bound on Lipschitz extension moduli was recently proved by Naor [15]. Roughly speaking, he showed that in any finite-dimensional Banach space, there are subsets with bad Lipschitz extension properties. More precisely, there is a universal constant c > 0 such that the following holds. In every n -dimensional Banach space X , there exist a subset A ⊂ X and a 1-Lipschitz map f : A → Y to a Banach space Y , such that every Lipschitz extension F : X → Y of f has Lipschitz constant at least n c .

Many other Lipschitz extension results can be found in the books [16–19] and for some results in the linear category (see, e.g., [20, Section III.B] or [21, Section 8]). For an introduction to extension results in topology, I strongly recommend [22].

1.2 Spaces of finite Nagata dimension

In this article, we will mostly consider arbitrary domain spaces X , but we will restrict ourselves to subsets A ⊂ X that are finite-dimensional in a quantifiable metric sense. A few definitions are in order. A covering of a metric space is said to have s -multiplicity at most n if every subset of the space of diameter less than s meets at most n members of the covering. We emphasize that throughout this article, we will always consider coverings by arbitrary subsets. The following definition is a variant of Gromov’s asymptotic dimension, taking into account both large- and small-scale properties of the metric space. Let n be a non-negative integer and c a positive real number.

This definition is due to Assouad [23], building on earlier work of Nagata [24]. Variants of Definition 1.1, where only small or large scales are considered, are called linearly controlled metric dimension and Higson property, respectively (see [25,26] for the precise definitions and further properties).

The smallest n ≥ 0 such that X satisfies Nagata ( n , c ) for some c is denoted by dim N ( X ) and called the Nagata dimension of X . The topological dimension dim ( X ) of X is at most dim N ( X ) (see [2, Theorem 2.2]) and this inequality can be strict in general, as for example, dim ( Z ) = 0 , but dim N ( Z ) = 1 . More dramatically, there are even finitely generated groups that have infinite Nagata dimension [27]. On the other hand, examples of metric spaces X with dim ( X ) = dim N ( X ) include compact connected Riemannian manifolds, Euclidean buildings of finite rank, uniformly disconnected spaces in the sense of David and Semmes, and Carnot groups equipped with the Carnot-Carathéodory distance [2,28,29].

A Hadamard manifold is a complete simply connected Riemannian manifold of non-positive sectional curvature. If the sectional curvature of a Hadamard manifold M is negatively pinched, then dim N ( M ) = dim ( M ) (see [2, Theorem 3.7]). Moreover, there exists a universal constant c such that every Hadamard manifold of dimension n = 2 , 3 satisfies Nagata ( n , c ) (see [30, Theorem 5.7.3] and [31, Theorem 3]). We remark that it is an open question whether all Hadamard manifolds have finite Nagata dimension or not.

Another important class of examples is obtained by considering minor-closed families of graphs. Very recently, building on a breakthrough result of Bonamy et al. [32] for the asymptotic dimension, Distel [33] and Liu [34] proved the following result. Let ℱ be a minor-closed family of graphs not containing every finite graph. Then, there exists c = c ( ℱ ) such that every member of ℱ satisfies Nagata ( 2 , c ) . Here, graphs are metrized in the usual way by equipping them with the shortest-path distance.

Many results from classical dimension theory also apply to the Nagata dimension. For example, the product formula dim N ( X × Y ) ≤ dim N ( X ) + dim N ( Y ) holds and also a Hurewicz-type theorem for Lipschitz maps [35]. Moreover, analogous to the topological dimension, Nagata dimension can be characterized in terms of extension properties of maps to spheres. More precisely, Brodskiy et al. [36] showed that if X is a metric space of finite Nagata dimension, then dim N ( X ) ≤ n if and only if the pair ( X , S n ) has the Lipschitz extension property (see, e.g., [37, Theorem 1.9.3] for analogous conditions for dim ( X ) ≤ n ).

1.3 Lipschitz n -connected spaces

We now proceed by introducing the type of target spaces considered by Lang and Schlichenmaier [2]. Recall that a topological space is said to be n -connected if for every 0 ≤ m ≤ n , any continuous map from the m -sphere to the space admits a continuous extension to the ( m + 1 ) -ball. Lang and Schlichenmaier adapted this definition to the metric setting as follows.

For example, the sphere S n is Lipschitz ( n − 1 ) -connected (see [30, Proposition 6.2.6]), but clearly not Lipschitz n -connected by Brouwer’s theorem. On the other hand, the unit sphere in an infinite-dimensional Banach space is Lipschitz n -connected for every n . In fact, Benyamini and Sternfeld [38] proved that these spheres are Lipschitz contractible and it is easy to check that any Lipschitz contractible space satisfies LC ( n , λ ) for every n with a uniform constant λ . In Section 7, we put some effort into finding optimal values of λ . For example, we show that Banach spaces satisfy LC ( n , 3 ) for every n . We will prove this statement more generally for metric spaces admitting conical bicombings (see Proposition 7.3). Thus, for example, it is also valid for all Hadamard manifolds.

There are also many examples of metric spaces satisfying LC ( n , λ ) , which do not admit conical bicombings. By results of Lang-Schlichenmaier [2, Theorem 5.1] and Wenger and Young [39, Theorem 1.1], other examples of Lipschitz n -connected spaces are n -connected closed Riemannian manifolds, and Carnot groups that can be realized as a jet space Carnot group J s ( R n + 1 ) . Also, certain horospheres in symmetric spaces of noncompact type of rank n ≥ 2 are Lipschitz ( n − 2 ) -connected (see [40,41]).

1.4 Lang and Schlichenmaier’s theorem

In view of the many examples stated earlier, the following explicit version of Lang and Schlichenmaier’s extension theorem [2, Theorem 1.6] seems to cover many of the possible settings in which one might wish to construct Lipschitz extensions.

The reader may be surprised at the appearance of the exponential tower 1 0 1 0 10 in (1.1). As it turns out, this term is only an artifact of the proof. It arises because in an intermediate step to (1.1), we encounter a term of the form ( C ⋅ n ) n which we decided to estimate by a large multiple of n n , so that (1.1) does not consist of too many terms. In fact, (1.1) can be replaced by the slightly more complicated bound

Depending on the value of n , one might prefer (1.2) over (1.1) or vice versa. We remark that in the special case, when n = 0 , a much better bound than (1.2) was obtained in the study by Basso and Sidler [42]. There, it is shown that if n = 0 , then f admits an 8 c λ -Lipschitz extension (which factors through a metric R -tree).

Our proof of Theorem 1.3 closely follows Lang and Schlichenmaier’s original proof. The main idea is to construct an extension F : X → Y such that F restricted to X \ A factors through a simplicial complex Σ of dimension at most n + 1 . This idea has its roots in classical topology where it is used for example in the proof of the Kuratowski and Dugundji extension theorem [43,44]. The complex Σ will be the nerve complex of a covering of X \ A . This covering has properties that mimic the classical Whitney cube decomposition of an open set in Euclidean space. The map F is then obtained by successive extensions of a certain map defined on the 0-skeleton of Σ . The constant ≈ λ n ⋅ n n appearing in (1.1) is due to the fact that we need to apply LC ( n , λ ) a total of ( n + 1 ) -times to inductively extend this map to the ( n + 1 ) -skeleton of Σ , which agrees with Σ .

1.5 Target spaces of generalized non-positive curvature

For many target spaces of interest such as Banach spaces or Hadamard manifolds, the aforementioned inductive application of LC ( n , λ ) can be avoided. The main reason for this is that (informally speaking) such spaces admit “barycentric coordinates.” By means of these coordinates, Lipschitz maps defined on the vertices of a simplicial complex can be extended to the entire complex in a single step. For example, if Y is a Banach space and f : Σ ( 0 ) → Y is defined on the vertices Σ ( 0 ) of a pure n -dimensional simplicial complex Σ , then F : Σ → Y , which on each n -simplex Δ ⊂ Σ is defined by

is an extension of f . Moreover, this extension has the same Lipschitz constant as f if Σ ⊂ R I is equipped with the metric induced by the ℓ 1 -norm on R I .

Using a barycenter construction going back to Es-Sahib and Heinich [45], one can show that any Y satisfying the following definition admits “barycentric coordinates” and thus extensions as earlier exist for such target spaces.

It follows directly from the definition that m ( x , y ) is a metric midpoint of x and y . In particular, any complete gNPC space is a geodesic metric space (see [46, p. 4] for the definition). We remark that for midpoints in Euclidean space, the inequality in (1.3) becomes an equality. Therefore, for complete metric spaces, (1.3) can be interpreted as follows. For any three points x , y , and z in Y , there is a geodesic triangle in Y with vertices x , y , and z such that the distances between the midpoints of the sides are not greater than the corresponding distances between the midpoints in the Euclidean comparison triangle. It may be helpful to compare this with the definition of a CAT ( 0 ) space, where it is required that all geodesic triangles in the space are not thicker than their comparison triangles in the Euclidean plane.

Examples of spaces of generalized non-positive curvature are Banach spaces (with m ( x , y ) = 1 2 ( x + y ) ) and CAT ( 0 ) spaces, where m ( x , y ) denotes the unique midpoint of x and y . Additional examples are Busemann spaces [47] and injective metric spaces [48]. On the other hand, complete gNPC spaces are contractible, and thus, the circle S 1 is not a gNPC space. A complete characterization of gNPC spaces as certain “convex” subsets of injective metric spaces can be found in [49]. The class of gNPC spaces enjoys many good structural properties. For example, it is closed under Gromov-Hausdorff limits, ℓ p -products for p ∈ [ 1 , ∞ ] , and also 1-Lipschitz retractions.

For target spaces of generalized non-positive curvature, the upper bound on the Lipschitz constant in Theorem 1.3 can be improved considerably.

A version of the theorem with (1.4) replaced by O ( c ⋅ n 3 ) was proved by Naor and Silberman [50, Corollary 5.2]. Their result resolves a conjecture of Brudnyi and Brudnyi [17, p. 48], who asked whether, for spaces A ⊂ X satisfying Nagata ( n , c ) and CAT ( 0 ) target spaces Y , it is always possible to extend 1-Lipschitz maps A → Y to Lipschitz maps X → Y , whose Lipschitz constants are bounded by K ⋅ n a , for some K , a > 0 depending only on the Nagata constant c . The sublinear dependence of (1.4) on the parameters n and c shows that Lipschitz extensions in this setting have even better properties than what was generally expected.

The main idea behind the proofs of Theorems 1.3 and 1.5 is the same. In particular, the restriction of F to X \ A thus factorizes by construction through a simplicial complex Σ , which is the nerve complex of a Whitney-type covering of X \ A . The appearance of log 2 ( n + 2 ) in (1.4) stems from the fact that we use a method of Johnson et al. [13] to construct particularly good Lipschitz partitions of unity. The main difference between the proofs is that, as mentioned earlier, we use “barycentric coordinates” to extend maps from Σ ( 0 ) to Y in one single step. More precisely, by a result of Descombes [51], we know that spaces of generalized non-positive curvature Y admit a barycenter map β : P 1 ( Y ) → Y in the sense of Sturm [52, Remark 6.4]. Here, P 1 ( Y ) denotes the space of Radon probability measures on Y with a bounded first moment. Using such a barycenter map β , one can extend any f : Σ ( 0 ) → Y to a map F : Σ → Y by setting

on each n -simplex Δ ⊂ Σ . Since β is 1-Lipschitz with respect to the 1-Wasserstein distance, this extension has the desired properties. The necessary background on Wasserstein distances is recalled in Section 2.4.

For a given domain space A , for which it is known from general considerations that it satisfies Nagata ( n , c ) for some c , it is often very difficult to obtain a good estimate on c . For example, already, the asymptotic behavior of the smallest c such that R n satisfies Nagata ( n , c ) seems to be unknown. Other instances where n is known but the estimation of c is difficult are closed Riemannian n -manifolds. For such spaces, the following Lipschitz extension result sometimes yields better bounds than Theorem 1.5.

The proof of the theorem is quite short and relies on straightforward properties of simplicial complexes and the generalized Kirszbraun theorem of Lang and Schroeder [9].

1.6 Extension results of Lee and Naor

We now turn our attention to some of the results of Lee and Naor [1]. A metric space X is called M -doubling if every ball in X can be covered by at most M balls of half the radius. It follows directly from [2, Lemma 2.3] that any M -doubling metric spaces satisfies Nagata ( M 3 , 2 ) . Thus, all the Lipschitz extension theorems mentioned earlier apply specifically to doubling metric spaces. In particular, the following explicit version of [1, Theorem 1.6] is a direct consequence of Theorem 1.5.

A streamlined proof of this theorem has already been given by Brudnyi and Brudnyi [53]. Their proof is constructive and uses a Lipschitz extension operator based on the integral average over balls. However, the obtained universal constant is not specified, and it probably requires some work to make it explicit. A generalization of Theorem 1.7 to spaces of pointwise homogeneous type, which include, for example, the standard hyperbolic spaces H n , can be found in [54] and [17, Section 7]. It follows from Theorem 1.7 that if A is a finite metric space that consists of at most n points, then any 1-Lipschitz map f : A → Y admits a

Lipschitz extension F : X → Y . A deep theorem of Lee and Naor [1] improved the asymptotics of this bound. We have the following explicit version of [1, Theorem 1.10].

Lee and Naor’s proof uses an elegant multiscale extension argument and relies on stochastic decomposition techniques from theoretical computer science. A rough sketch of their proof strategy can be found in Section 4, where we essentially follow this strategy and give a self-contained proof of the theorem. The proof crucially relies on certain padded decompositions of finite metric spaces. We construct these decompositions in Lemma 2.3 using the methods of [55–57]. Contrary to what is common in the literature, our formulation of the lemma does not use any stochastic language. Moreover, our proof can be viewed as a de-randomized version of the usual proof. However, it is mathematically equivalent to the usual stochastic proof, and therefore, the whole reformulation is essentially just a matter of linguistics. Theorem 1.8 admits a concise reformulation using the absolute extendability constant of a metric space.

Hence, the aforementioned theorem states that if X consists of at most n points, then

For small n , this estimate can be further improved. It follows directly from [58, Theorem 1.4] and the Kadets-Snobar theorem [59] that

for any n -point metric space X . This estimate is stronger than (1.6) for any n ≤ 1 0 7 . Let us remark here that little is known about the precise values of æ ( n ) = sup { æ ( X ) : ∣ X ∣ = n } for small n . It is conceivable that good bounds of these constants for small n could shed some light on the general behavior of the function æ ( n ) .

1.7 Structure of the article

This article is organized as follows. In Section 2, we prove two lemmas relating pointwise Lipschitz constants to (global) Lipschitz constants. We also recall the necessary background from optimal transport theory regarding our use of “barycentric coordinates.” Section 3 is devoted to the proof of Theorem 1.5. The proof of this theorem can be seen as a blueprint for the more elaborate proof of Theorem 1.3 in Section 8. In the next section, we prove Theorem 1.8, our explicit version of the extension theorem of Lee and Naor. The main purpose of Section 5 is to introduce terminology regarding simplicial complexes. Using some of the tools developed there, we also give the straightforward proof of Theorem 1.6. In Section 6, we establish a general extension theorem relating Lipschitz extension constants to parameters of certain coverings, which we call Whitney coverings. Section 7 deals with Lipschitz n -connected spaces. We discuss alternative definitions there and show in Proposition 7.3 that gNPC -spaces are Lipschitz n -connected for every n . Finally, in Section 8, we prove Theorem 1.3 by the use of Theorem 6.3.

2.1 Basic metric notions

Let X = ( X , d ) be a metric space. For any x ∈ X and r > 0 , we use B ( x , r ) to denote the closed ball in X with center x and radius r . Given two subsets A , B ⊂ X , we write

to denote the infimal distance between A and B . To ease notation, we write d ( x , B ) for d ( { x } , B ) . We use

to denote the open r -neighborhood of B in X . A covering of X is, by definition, a collection ( B i ) i ∈ I of subsets of X such that ⋃ i ∈ I B i = X . For all coverings constructed in this article, we tacitly assume that they consist only of non-empty subsets. This does not cause any problems because whenever X is non-empty and the empty set is removed from a covering of X , the resulting collection is still a covering.

2.2 Pointwise Lipschitz constants

For any map f : X → Y between metric spaces, we denote by

the (upper) pointwise Lipschitz constant of f at x . The following lemma shows that in a quasiconvex domain space, one can prove the Lipschitz continuity by establishing an upper bound on the pointwise Lipschitz constants. Recall that X is c -quasiconvex if any two points in X can be joined by a continuous path of length at most c times their distance.

The object of study of this article are extensions F : X → Y of a given Lipschitz map f : A → Y . Since we consider arbitrary domain spaces, by Kuratowski’s embedding, there is no loss in generality to assume that X is a Banach space (and thus, in particular, 1-quasiconvex). Hence, by the aforementioned lemma, to check whether F is Lipschitz, it suffices to bound the pointwise Lipschitz constant of F . Our next lemma simplifies this even further and shows that for continuous F , it is sufficient to only bound the pointwise Lipschitz constants of points in X \ A . This lemma is very useful and is applied in all the proofs of our main theorems.

2.3 Good coverings

Given a covering of a metric space, one might ask how often a given point is contained well within the interior of the members of the covering. For a covering ( B i ) i ∈ I , a fixed point x ∈ X , and D > 0 , the ratio

tells us the percentage of the sets from our collection that contain x in a way that it is at least distance D away from their boundaries. For our Lipschitz extension theorems, it turns out to be helpful to work with coverings where this ratio is as large as possible. We informally refer to such coverings as good coverings. For example, spaces that satisfy Nagata ( n , c ) admit good coverings. More precisely, such spaces X admit for any D > 0 a 2 c D -bounded covering such that δ x ( D ) ≥ 1 n + 1 for every x ∈ X . This can easily be seen by looking at the open ( D ⁄ 2 ) -neighborhoods of the members of the Nagata cover at scale D . This simple observation will be helpful in the proof of Theorem 1.5 in Section 3.

The proof of the Lee-Naor extension theorem in Section 4 requires the following more refined estimate on δ x ( D ) for finite metric spaces.

2.4 Spaces of generalized non-positive curvature

Let X be a metric space, and denote by ℬ ( X ) its Borel σ -algebra. A signed measure λ : ℬ ( X ) → R is said to be a real Radon measure if its total variation μ = ∣ λ ∣ is an inner regular measure, i.e., μ ( B ) is equal to the supremum μ ( K ) over all compact subset K ⊂ B . We denote by ℳ ( X ) the vector space of all real Radon measures on X . Let ℳ s ( X ) be the linear subspace of ℳ ( X ) generated by all measures of the form δ x − δ y for x , y ∈ X . The following expression

clearly defines a norm on ℳ s ( X ) . This norm is called the Kantorovich-Rubinstein norm, and the completion of ℳ s ( X ) with respect to ‖ ⋅ ‖ KR is called the Lipschitz free space over X and is denoted by ℱ ( X ) . If ( X , p ) is a pointed metric space, then δ : X → ℱ ( X ) defined by δ ( x ) = δ x − δ p is an isometric embedding. For Banach spaces, this map is not linear, but it admits a bounded linear inverse β : ℱ ( X ) → X , which satisfies β ( λ ) = ∫ x d λ for all λ ∈ ℳ s ( X ) . This barycenter map β is a key notion in the theory of Lipschitz free spaces in Banach space theory [63].

In the following, we discuss the existence of such barycenter maps in the metric setting. Let P 1 ( X ) denote the set of all Radon probability measures μ : ℬ ( X ) → [ 0 , 1 ] on X such that

for some x 0 ∈ X (and thus also for any other point in X ). Note that if μ , ν ∈ P 1 ( X ) , then ‖ μ − ν ‖ KR < ∞ , and thus,

defines a metric on P 1 ( X ) . This metric is mostly referred to as 1-Wasserstein distance. However, the history of this distance (see, e.g., [64, pp. 106–108]) shows that a more neutral name would be more appropriate, as it was repeatedly (re-)discovered by many different authors. Throughout this article, P 1 ( X ) will always be endowed with W 1 . Usually, the 1-Wasserstein distance is defined in terms of optimal couplings of probability measures, and (2.7) is then a consequence of the Kantorovich-Rubinstein duality theorem. However, since we will have no use of this particular viewpoint, we do not discuss this further. For more information on the Kantorovich-Rubinstein duality theorem, we refer the reader to [65]. The following definition is due to Sturm [52, Remark 6.4].

It is easy to check that δ : X → P 1 ( X ) defined by x ↦ δ x is an isometric embedding. Thus, the existence of a barycenter map β : P 1 ( X ) → X is equivalent to saying that δ admits a 1-Lipschitz continuous left inverse. The following result is the cumulation of the works of Es-Sahib and Heinich [45], Navas [66], and Descombes [51].

We refer to [49, Section 2] for a justification of how this result follows directly from [51, Theorem 2.5] (see also [67, Section 3.2]). Other barycentric constructions can be found in [68,69]. For our proofs, it would actually be suffice to just have a barycenter map β : P f ( X ) → X defined on the subset P f ( X ) ⊂ P 1 ( X ) of all μ ∈ P 1 ( X ) whose support is finite. Distances between such measures can be estimated as follows.

We remark that if μ and ν denote the measures on the left-hand side of (2.8), the aforementioned sum is equal to the total variation ∣ μ − ν ∣ ( X ) .